Nombres qui
pavent le plan
I] GÉNÉRALITÉS
Soit un plan infini couvert de cases 1 x 1 initialement vides.
Loi 1. Une case vide peut se remplir au maximum d’un seul chiffre d.
Loi 2. Est appelé « voisinage » du chiffre d le carré 3 x 3 dont d occupe le centre :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | d | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Loi 3. Si un chiffre d occupe une case, il doit y avoir exactement d exemplaires de d dans son voisinage.
Exemple : il y a ci-dessous 5 exemplaires du chiffre 5 (gras) dans le voisinage du chiffre 5 (gras) :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | 5 | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | 5 | 5 | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Notons déjà qu’un plan (infini) traversé par une double diagonale (infinie) de 5 obéit aux lois de voisinage ci-dessus :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | 5 | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | 5 | 5 | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | 5 | 5 | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | 5 | 5 | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | 5 | 5 | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | 5 | 5 | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | 5 | 5 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Il en est de même ici où l’on ajoute une diagonale simple de 3 :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 3 | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 3 | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | 5 | 3 | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | 5 | 5 | 3 | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | 5 | 5 | 3 | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | 5 | 5 | 3 | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | 5 | 5 | 3 | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | 5 | 5 | 3 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | 5 | 5 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
... et là, avec l’« escalier » 1-2-2 :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 3 | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 3 | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 5 | 5 | 3 | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 2 | 5 | 5 | 3 | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | 1 | 5 | 5 | 3 | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | 2 | 2 | 5 | 5 | 3 | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | 1 | 5 | 5 | 3 | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | 2 | 2 | 5 | 5 | 3 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | 1 | 5 | 5 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | 2 | 2 | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
... ou là :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | | 5 | | 5 | | 5 | | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | | 5 | | 5 | | 5 | | 5 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | | 5 | | 5 | | 5 | | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | | 5 | | 5 | | 5 | | 5 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | | 5 | | 5 | | 5 | | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | | 5 | | 5 | | 5 | | 5 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | | 5 | | 5 | | 5 | | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | | 5 | | 5 | | 5 | | 5 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | | 5 | | 5 | | 5 | | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | | 5 | | 5 | | 5 | | 5 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Voici un plan dont toutes les cases sont occupées. Le chiffre 9 pave ce plan :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Y a-t-il d’autres nombres qui pavent le plan, comme 9 ? Nous y reviendrons plus bas.
II] LES RECTANGLES
dans R1 et R2 & LES BOÎTES dans R3
Tous les rectangles ci-dessous sont pavés de cases 1 x 1 identiques portant un chiffre – et ce chiffre ne peut être supérieur à 4 :
R1) Les rectangles 1 x n.
n = 1, une seule solution :
+---+
| 1 |
+---+
n = 2, une seule solution :
+---+---+
| 2 | 2 |
+---+---+
n = 3, une seule solution (aux rotations et symétries près) :
+---+---+---+
| 1 | 2 | 2 |
+---+---+---+
n = 4, une seule solution :
+---+---+---+---+
| 1 | 2 | 2 | 1 |
+---+---+---+---+
n = 5, une seule solution :
+---+---+---+---+---+
| 2 | 2 | 1 | 2 | 2 |
+---+---+---+---+---+
n = 6, une seule solution (aux rotations et symétries près) :
+---+---+---+---+---+---+
| 1 | 2 | 2 | 1 | 2 | 2 |
+---+---+---+---+---+---+
Etc. On a compris le (peu intéressant) principe de remplissage des rectangles 1 x n.
R2) Les rectangles 2 x n.
n = 1, voir ci-dessus.
n = 2, deux solutions :
+---+---+ +---+---+
| 1 | 3 | | 4 | 4 |
+---+---+ +---+---+
| 3 | 3 | | 4 | 4 |
+---+---+ +---+---+
n = 3, cinq solutions :
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 3 | 3 | | 1 | 2 | 3 | | 1 | 3 | 2 |
+---+---+---+ +---+---+---+ +---+---+---+
| 2 | 2 | 3 | | 2 | 3 | 3 | | 3 | 3 | 2 |
+---+---+---+ +---+---+---+ +---+---+---+
+---+---+---+ +---+---+---+
| 3 | 1 | 2 | | 2 | 4 | 4 |
+---+---+---+ +---+---+---+
| 3 | 3 | 2 | | 2 | 4 | 4 |
+---+---+---+ +---+---+---+
n = 4, six solutions :
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 1 | 2 | 3 | 2 | | 1 | 3 | 3 | 2 | | 1 | 3 | 4 | 4 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 2 | 3 | 3 | 2 | | 2 | 2 | 3 | 2 | | 3 | 3 | 4 | 4 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 2 | 1 | 3 | 2 | | 3 | 1 | 4 | 4 | | 2 | 4 | 4 | 2 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 2 | 3 | 3 | 2 | | 3 | 3 | 4 | 4 | | 2 | 4 | 4 | 2 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
n = 5, vingt-et-une solutions :
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 1 | 2 | 3 | 4 | 4 | | 1 | 3 | 3 | 4 | 4 | | 1 | 3 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 2 | 3 | 3 | 4 | 4 | | 2 | 2 | 3 | 4 | 4 | | 3 | 3 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 1 | 3 | 4 | 4 | 2 | | 2 | 1 | 3 | 4 | 4 | | 3 | 1 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 3 | 4 | 4 | 2 | | 2 | 3 | 3 | 4 | 4 | | 3 | 3 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 1 | 4 | 4 | 2 | | 2 | 3 | 1 | 4 | 4 | | 3 | 3 | 1 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 3 | 4 | 4 | 2 | | 2 | 3 | 3 | 4 | 4 | | 3 | 2 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 2 | 1 | 4 | 4 | | 1 | 3 | 1 | 2 | 3 | | 1 | 3 | 1 | 3 | 3 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 3 | 2 | 4 | 4 | | 3 | 3 | 2 | 3 | 3 | | 3 | 3 | 2 | 2 | 3 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 1 | 3 | 2 | 3 | 3 | | 1 | 3 | 2 | 2 | 3 | | 1 | 3 | 2 | 1 | 3 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 3 | 1 | 2 | 3 | | 3 | 3 | 1 | 3 | 3 | | 3 | 3 | 2 | 3 | 3 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 1 | 3 | 2 | 3 | 3 | | 1 | 3 | 2 | 3 | 1 | | 1 | 3 | 2 | 3 | 3 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 3 | 2 | 1 | 3 | | 3 | 3 | 2 | 3 | 3 | | 3 | 3 | 2 | 3 | 1 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 1 | 2 | 1 | 3 | | 3 | 1 | 2 | 3 | 3 | | 4 | 4 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
| 3 | 3 | 2 | 3 | 3 | | 3 | 3 | 2 | 1 | 3 | | 4 | 4 | 2 | 4 | 4 |
+---+---+---+---+---+ +---+---+---+---+---+ +---+---+---+---+---+
Lars Blomberg a creusé le problème et précise :
For
2 x n rectangles I made a plot showing the exponential growth of the series (see picture
below).
I
also discovered the following.
A 2 x n
solution can be split whenever there is not the same digit on either side of
the split
Example
1:
132314431244 13 2 31 44 31 2 44
332334433244 33 2 33 44 33 2 44
Example
2:
232233244133 2 322 33 2 44 133
233131244223 2 331 31 2 44 223
It seems
that all 2 x n can be built from a limited number of
parts as follows:
Parts of
width 1
2
2
Parts of
width 2
13 31 33 33 44
33 33 13 31 44
Parts of
width 3
123 133
223 233 321 322 331 332
233 223
133 123 332 331 322 321
This has
been verified up to size 2 x 15.
Using
this approach it would be possible to generate 2 x n pavings
for large n with greater speed, although all solutions must still be kept in
memory for duplicate checking.
The same
approach for 3 x n works only partially.
There are
some solutions that cannot be split at all, for example:
1333333331
3221221223
1333333331
or
1332244331
4431344344
4422332244
R2) Les rectangles 3 x n.
n = 1 et n = 2, voir supra.
n = 3, cinq solutions :
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 2 | 2 | | 1 | 4 | 4 | | 1 | 3 | 1 |
+---+---+---+ +---+---+---+ +---+---+---+
| 3 | 3 | 1 | | 3 | 4 | 4 | | 3 | 2 | 3 |
+---+---+---+ +---+---+---+ +---+---+---+
| 3 | 2 | 2 | | 3 | 3 | 1 | | 1 | 3 | 2 |
+---+---+---+ +---+---+---+ +---+---+---+
+---+---+---+ +---+---+---+
| 1 | 2 | 1 | | 1 | 2 | 1 |
+---+---+---+ +---+---+---+
| 2 | 4 | 4 | | 2 | 3 | 3 |
+---+---+---+ +---+---+---+
| 1 | 4 | 4 | | 1 | 3 | 1 |
+---+---+---+ +---+---+---+
n = 4, dix-sept solutions trouvées par Lars (en ignorant rotations et symétries) :
----
1212
2442
1441
----
1221
3344
1344
----
1221
3344
3144
----
1223
4433
4422
----
1233
2443
1441
----
1244
2344
1331
----
1244
2344
3322
----
1322
2331
2122
----
1322
3344
2244
----
1331
2344
2144
----
1331
3223
1331
----
1441
2443
2133
----
2122
2344
3344
----
2144
2344
3322
----
2213
4433
4422
----
2233
4413
4422
----
2233
4431
4422
Voici un tableau très clair de Lars montrant le remplissage de rectangles de plus en plus grands :
R3) Les boîtes n x n x n.
Dans l’espace R3 le voisinage d’un chiffre d est représenté par le cube 3 x 3 x 3 centré sur d. Pour la boîte 2 x 2 x 2 ci-dessous, chaque cellule individuelle « touche » donc les 7 autres (par une face, une arête ou un sommet). Nous avons illustré la boîte 2 x 2 x 2 en disposant ses 2 niveaux côte-à-côte.
n = 1, voir supra.
n = 2, vingt-quatre solutions (trouvées par Olivier Miakinen, merci à lui !) :
+---+---+ +---+---+ +---+---+ +---+---+
| 8 | 8 | | 8 | 8 | | 7 | 7 | | 7 | 7 |
+---+---+ & +---+---+ +---+---+ & +---+---+
| 8 | 8 | | 8 | 8 | | 7 | 7 | | 7 | 1 |
+---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| 6 | 6 | | 6 | 2 | | 6 | 6 | | 6 | 2 | | 6 | 6 | | 6 | 2 |
+---+---+ & +---+---+ +---+---+ & +---+---+ +---+---+ & +---+---+
| 6 | 6 | | 6 | 2 | | 6 | 6 | | 2 | 6 | | 2 | 6 | | 6 | 6 |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| 5 | 5 | | 5 | 3 | | 5 | 5 | | 5 | 3 | | 3 | 5 | | 5 | 3 |
+---+---+ & +---+---+ +---+---+ & +---+---+ +---+---+ & +---+---+
| 5 | 5 | | 3 | 3 | | 3 | 5 | | 5 | 3 | | 5 | 5 | | 3 | 5 |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+
| 5 | 5 | | 5 | 2 | | 5 | 5 | | 5 | 2 |
+---+---+ & +---+---+ +---+---+ & +---+---+
| 5 | 5 | | 1 | 2 | | 5 | 5 | | 2 | 1 |
+---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| 5 | 5 | | 5 | 2 | | 5 | 5 | | 5 | 2 | | 5 | 5 | | 5 | 1 |
+---+---+ & +---+---+ +---+---+ & +---+---+ +---+---+ & +---+---+
| 1 | 5 | | 5 | 2 | | 2 | 5 | | 5 | 1 | | 2 | 5 | | 5 | 2 |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+
| 1 | 5 | | 5 | 2 |
+---+---+ & +---+---+
| 5 | 5 | | 2 | 5 |
+---+---+ +---+---+
+---+---+ +---+---+
| 4 | 4 | | 1 | 3 |
+---+---+ & +---+---+
| 4 | 4 | | 3 | 3 |
+---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| 1 | 4 | | 4 | 3 | | 4 | 1 | | 4 | 3 | | 4 | 4 | | 4 | 3 |
+---+---+ & +---+---+ +---+---+ & +---+---+ +---+---+ & +---+---+
| 4 | 4 | | 3 | 3 | | 4 | 4 | | 3 | 3 | | 4 | 1 | | 3 | 3 |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| 1 | 4 | | 4 | 3 | | 4 | 1 | | 4 | 3 | | 4 | 4 | | 4 | 3 |
+---+---+ & +---+---+ +---+---+ & +---+---+ +---+---+ & +---+---+
| 3 | 4 | | 4 | 3 | | 3 | 4 | | 4 | 3 | | 3 | 1 | | 4 | 3 |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
| 3 | 4 | | 1 | 3 | | 3 | 4 | | 4 | 3 | | 3 | 4 | | 4 | 3 |
+---+---+ & +---+---+ +---+---+ & +---+---+ +---+---+ & +---+---+
| 4 | 4 | | 3 | 4 | | 4 | 1 | | 3 | 4 | | 4 | 4 | | 3 | 1 |
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+
n = 3 : le nombre de solutions est hors de portée des crayon, papier, cerveau du signataire... Nous avons illustré ci-dessous une boîte 3 x 3 x 3 parmi d’autres. Mais parmi combien d’autres possibles ? Des centaines ? Le dénombrement des boîtes de dimension k x l x m est une question ouverte – comme la question, soulevée par Olivier Miakinen, de trouver le plus grand nombre que puisse contenir une boîte aux mesures k x l x m finies...
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 6 | 6 | | 2 | 6 | 3 | | 2 | 3 | 3 |
+---+---+---+ +---+---+---+ +---+---+---+
| 4 | 6 | 6 | | 4 | 6 | 5 | | 1 | 5 | 5 |
+---+---+---+ +---+---+---+ +---+---+---+
| 4 | 3 | 3 | | 4 | 3 | 5 | | 2 | 2 | 5 |
+---+---+---+ +---+---+---+ +---+---+---+
III] RUBANS
INFINIS dans R1, R2 et R3
Un ruban dans R1 n’est rien d’autre qu’une ligne infinie. Il n’y a que deux manières de paver une telle ligne :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
et :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
Les rubans dans R2 sont de deux types : horizontaux et obliques.
Voici quelques rubans horizontaux simples à 2 couches :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 3 | 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3 | 1 | 3 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 1 | 2 | 2 | 3 | 3 | 3 | 1 | 2 | 2 | 3 | 3 | 3 | 1 | 2 | 2 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 3 | 3 | 3 | 1 | 2 | 2 | 3 | 3 | 3 | 1 | 2 | 2 | 3 | 3 | 3 | 1 | 2 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 3 | 2 | 2 | 4 | 4 | 3 | 2 | 2 | 4 | 4 | 3 | 2 | 2 | 4 | 4 | 3 | 2 | 2 | 4 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 3 | 3 | 1 | 4 | 4 | 3 | 3 | 1 | 4 | 4 | 3 | 3 | 1 | 4 | 4 | 3 | 3 | 1 | 4 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 | 4 | 4 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 1 | 5 | 5 | 1 | 5 | 5 | 1 | 5 | 5 | 1 | 5 | 5 | 1 | 5 | 5 | 1 | 5 | 5 | 1 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 5 | 5 | 5 | 1 | 5 | 5 | 5 | 5 | 5 | 1 | 5 | 5 | 5 | 5 | 5 | 1 | 5 | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 1 | 5 | 5 | 5 | 5 | 5 | 1 | 5 | 5 | 5 | 5 | 5 | 1 | 5 | 5 | 5 | 5 | 5 | 1 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
Tous ces rubans présentent des pavages périodiques mais il est très simple d’en produire sans cette propriété :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 3 | 3 | 4 | 4 | 3 | 3 | 4 | 4 | 3 | 1 | 4 | 4 | 3 | 1 | 4 | 4 | 3 | 3 | 4 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 4 | 3 | 1 | 4 | 4 | 1 | 3 | 4 | 4 | 3 | 3 | 4 | 4 | 3 | 3 | 4 | 4 | 1 | 3 | 4 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
... les blocs jaunes/gris composées de 1 et de 3 peuvent en effet être orientés comme on le souhaite car le critère général de voisinage sera toujours respecté dans le ruban.
Voici quelques rubans horizontaux à trois couches :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 3 | 3 | 1 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 3 | 5 | 5 | 3 | 5 | 5 | 3 | 5 | 5 | 3 | 5 | 5 | 3 | 5 | 5 | 3 | 5 | 5 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
... etc. On comprend qu’il est facile de construire de tels rubans en superposant des rubans plus élémentaires.
Le plan ci-dessous, présenté tout au début de cette page, montre trois rubans : celui à une seule couche (composé des 3 jaunes), celui à deux couches (avec les 5 gras), et celui à trois couches (la juxtaposition des deux premiers) :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 3 | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 5 | 5 | 3 | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | 5 | 5 | 3 | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | 5 | 5 | 3 | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | 5 | 5 | 3 | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | 5 | 5 | 3 | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | 5 | 5 | 3 | | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | 5 | 5 | 3 | |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | 5 | 5 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | 5 | 5 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Tous les rubans obliques ne sont pas à 45° :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 4 | 2 | 4 | 4 | 4 | 2 | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | 4 | 2 | 4 | 4 | 4 | 2 | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | 4 | 2 | 4 | 4 | 4 | 2 | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | 4 | 2 | 4 | 4 | 4 | 2 | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | 4 | 2 | 4 | 4 | 4 | 2 | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | 4 | 2 | 4 | 4 | 4 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | 4 | 2 | 4 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
Nous n’avons pas exploré les rubans infinis dans R3 – à considérer plutôt comme barreaux pleins, pilastres ou poutrelles obliques –, car trop difficiles à visualiser (pour nous)...
Arrivés à ce stade de notre exploration, nous nous sommes intéressés aux pavages du plan R2 – et plus précisément aux nombres paveurs – lesquels seront expliqués dans un instant.
Montrons tout d’abord qu’il est aisé de paver le plan R2 en assemblant des rubans élémentaires – pavages qui peuvent être, comme nous l’avons vu, périodiques ou non. Celui-ci est périodique :
| | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | |
... celui-ci ne l’est pas – les couches « 122 » pouvant glisser les unes par rapport aux autres, à leur gré :
| | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | |
Qui dit pavage périodique dit tuile (ou tomette) ; voici, en jaune uni, la tuile qui pave un des plans rencontrés plus haut ; cette tuile s’inscrit dans un rectangle 2 x 3 :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2.| 1 | 2 | 2 | 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3.| 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2.| 1 | 2 | 2 | 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3.| 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Voici un autre pavage obtenu par empilement de rubans déjà rencontrés plus haut :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6.| 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6.| 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Ce pavage est réalisable par la tuile (verticale) 366 – tuile élémentaire par rapport à la tuile 122/333, car cette dernière comportait deux lignes de chiffres. Les tuiles élémentaires seront appelées désormais nombres paveurs (ou nombres qui pavent le plan) – et 366 est l’un d’eux. Remarquons que 366 peut paver le plan d’une infinité de manières – tournons d’abord le dessin précédent de 90° :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Remarquons ensuite que nous pouvons, à partir de cette position-ci, déplacer d’une case vers la gauche ou vers la droite n’importe quelles lignes sans altérer les propriétés du pavage. Nous avons déplacé ci-dessous la couche 2 d’une case vers la droite et les couches 4 et 8 d’une case vers la gauche : on vérifiera que la règle du voisinage n’est jamais enfreinte pour 3 ou 6 :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 1, n’ayant pas bougé
--+---+---+---+---\---+---+---\---+---+---+--
| 6 | 3 | 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | --> couche 2, une case à droite
--+---+---+---+---/---+---+---/---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 3, n’ayant pas bougé
--+---+---+---/---+---+---/---+---+---+---+--
| 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | <-- couche 4, une case à gauche
--+---+---+---\---+---+---\---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 5, n’ayant pas bougé
--+---+---+---+-|-+---+---+-|-+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 6, n’ayant pas bougé
--+---+---+---+-|-+---+---+-|-+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 7, n’ayant pas bougé
--+---+---+---/---+---+---/---+---+---+---+--
| 6 | 6 | 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | <-- couche 8, une case à gauche
--+---+---+---\---+---+---\---+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 9, n’ayant pas bougé
--+---+---+---+-|-+---+---+-|-+---+---+---+--
| 3 | 6 | 6.| 3 | 6 | 6 | 3 | 6 | 6.| 3 | couche 10, n’ayant pas bougé
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
Nous avons vu plus haut que 9 pave le plan. C’est la tuile la plus simple. On voit vite que 99 pave aussi le plan, ainsi que 999, 9999... La même technique de duplication peut être appliquée à 366 : les nombres 366366, 366366366... pavent aussi le plan. Et ce n’est pas tout : dès qu’un nombre pave le plan, toute permutation cyclique de ses chiffres produit un nombre qui pave aussi le plan. Ainsi 366 fait-il 636 et 663 qui pavent le plan – comme 636636 et 663663 dont les permutations cycliques des chiffres, à leur tour, pavent le plan, etc. Il y a donc une infinité de nombres paveurs. Mais comment trouver les plus petits d’entre eux ? Quel serait le début de la suite P des nombres qui pavent le plan ?
Revenons sur la tuile 122/333 et le pavage correspondant. Ce pavage est impossible à obtenir par un nombre paveur unique. En effet aucune ligne horizontale ou verticale ci-dessous n’affiche les chiffres 1, 2 et 3 à la fois :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2.| 1 | 2 | 2 | 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3.| 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2 | 1 | 2 | 2.| 1 | 2 | 2 | 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3.| 3 | 3 | 3 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 |
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
La technique du glissement permet pourtant de trouver un nombre paveur. On déplace d’un cran vers la gauche la ligne 3, puis de deux crans à gauche la ligne 5 – le nombre paveur unique 132323 apparaît (dans la première colonne) :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 | couche 1
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 | couche 2
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 2 | 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | <-- couche 3 (une case à gauche)
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3 | 3 | 3 | 3.| 3 | 3 | 3 | 3 | couche 4
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 2 | 2.| 1 | 2 | 2 | 1 | 2 | 2.| 1 | 2 | 2 | <-- couche 5 (deux cases à gauche)
--+---+---+---+---+---+---+---+---+---+---+--
| 3 | 3 | 3.| 3 | 3 | 3 | 3 | 3 | 3.| 3 | couche 6
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | .| | | | |
Certes 132323 pave-t-il un autre plan que celui de la tuile 122/333, mais au moins avons-nous trouvé un nombre paveur ! Voici le pavage par 132323, redressé à 90° – le motif jaune répété tient sur trois lignes et tous les critères de voisinage sont respectés :
| | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 | 2 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3.|
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 | 2 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3.|
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 | 2 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3.|
--+---+---+---+---+---+---+---+---+---+---+--
| 2 | 3 | 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| 1 | 3 | 2 | 3 | 2 | 3.| 1 | 3 | 2 | 3 |
--+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | |
35355 pave aussi le plan – dans une autre présentation graphique :
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
33555 pave le même plan que ci-dessus... mais « verticalement » (on a donc deux nombres différents qui pavent le même plan (33555 n’apparaît pas dans le cycle de 35355 – il est donc différent de ce dernier) :
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5
3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5
5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5
5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3
5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3 5 5 3 5 3
153535 pave le plan :
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5
5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1 5 3 5 3 5 1
Etc.
Le nombre 122356644355664435566 pave le plan ; il semble que ce soit le plus petit nombre paveur affichant six chiffres différents dans son écriture (il est impossible de faire figurer les chiffres 7, 8 ou 9 dans un tel nombre – explication après l’illustration) :
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
443556612235664435566443556612235664435566443556612235664435566443556612235664435566443556612235
443556644355661223566443556644355661223566443556644355661223566443556644355661223566443556644355
122356644355664435566122356644355664435566122356644355664435566122356644355664435566122356644355
On a vu qu’il y a moyen de cantonner les chiffres 6 dans une portion du plan, comme ci-dessus : un ruban infini composé de deux lignes de 6 :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
On voit vite qu’il est impossible de trouver un ruban qui puisse contenir des chiffres 7, quelle que soit la largeur (finie) de ce ruban. Les 7 prolifèrent vite, en effet, et si vous commencez au centre d’un (large) ruban, vous verrez qu’ils en atteignent vite un bord. Or les cases de bord d’un tel ruban n’ayant que 5 voisins, il faudrait aller chercher un dernier 7 en dehors de celui-ci (pour obéir au critère de voisinage) :
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 — 7 — 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---/-|-\---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
Impossible donc d’enfermer un 7 dans un ruban. Il en est de même pour le chiffre 8, bien sûr, dont la situation est encore « pire ». Quant à 9, on l’a vu, il colonise tout le plan à lui tout seul...
Quels seraient tous les pavages réguliers de R2 qui contiennent 7 ou 8 ? Il faut faire un peu de géométrie du carré 3 x 3, à l’origine du concept de voisinage. Si le 7 figure dans ce carré, c’est que 2 des 9 cases de son voisinage sont « libres ». Appelons a et b ces deux cases et cherchons comment paver le plan avec elles.
Examinons le cas où a et b se touchent par un bord, comme ci-dessous. On voit vite que les cases marquées d’une petite croix sont interdites à tout autre a ou b, sous peine de créer un voisinage contenant trois lettres, ce qui est interdit :
| | | | | | |
--+---+---+---+---+---+---+--
| | × | × | × | × | |
--+---+---+---+---+---+---+--
| | × | × | × | × | |
--+---+---+---+---+---+---+--
| | × | a | b | × | |
--+---+---+---+---+---+---+--
| | × | × | × | × | |
--+---+---+---+---+---+---+--
| | × | × | × | × | |
--+---+---+---+---+---+---+--
| | | | | | |
Essayons de paver le plan en répétant la configuration ci-dessus :
| | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | × | × | × | × | × | × | × | × | × | × | × | × | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | × | × | × | ▪ | ▪ | × | × | ▪ | ▪ | × | × | × | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | a | b | × | × | a | b | × | × | a | b | × | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | × | × | × | ▪ | ▪ | × | × | ▪ | ▪ | × | × | × | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | × | × | × | × | × | × | × | × | × | × | × | × | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | |
On voit qu’un 7 posé sur l’une des cases marquées d’un carré noir n’aura qu’un seule lettre dans son voisinage, au lieu de deux, ce qui est interdit. La configuration ci-dessous fonctionne en revanche – toutes les cases vides peuvent se remplir d’un 7 :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | | a | b | | a | b | | a | b | | a | b | | a | b | | a | b | | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
On obtient ce pavage (il suffit de remplacer a-b par 2-2 pour que tous les critères de voisinage soient respectés) :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Mais ce pavage n’est pas constructible par un nombre paveur. Il faut donc lui appliquer la technique des glissements successifs. Une première manière de glisser consiste à déplacer horizontalement chaque ligne a-b d’une case par rapport à la précédente :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Ça y est, le a jaune du bas est à la verticale de lui-même (le a souligné du haut). On remplace le couple a-b par le couple 2-2 et le nombre paveur 277777277 apparaît verticalement :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 | 2 | 7 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Voici le motif que dessine 277777277, redressé de 90° :
277777277277777277277777277277777277277777277277777277277777277277777277277
277277777277277777277277777277277777277277777277277777277277777277277777277
777277277777277277777277277777277277777277277777277277777277277777277277777
277777277277777277277777277277777277277777277277777277277777277277777277277
277277777277277777277277777277277777277277777277277777277277777277277777277
777277277777277277777277277777277277777277277777277277777277277777277277777
277777277277777277277777277277777277277777277277777277277777277277777277277
277277777277277777277277777277277777277277777277277777277277777277277777277
777277277777277277777277277777277277777277277777277277777277277777277277777
277777277277777277277777277277777277277777277277777277277777277277777277277
277277777277277777277277777277277777277277777277277777277277777277277777277
777277277777277277777277277777277277777277277777277277777277277777277277777
277777277277777277277777277277777277277777277277777277277777277277777277277
277277777277277777277277777277277777277277777277277777277277777277277777277
777277277777277277777277277777277277777277277777277277777277277777277277777
277777277277777277277777277277777277277777277277777277277777277277777277277
277277777277277777277277777277277777277277777277277777277277777277277777277
777277277777277277777277277777277277777277277777277277777277277777277277777
277777277 produira huit autres nombres paveurs par permutations cycliques de ses chiffres. Le plus petit d’entre eux est 277277777.
Une seconde manière de glisser les carrés 3 x 3 du diagramme 1 consiste à déplacer verticalement et ensemble les colonnes a-b (au lieu de déplacer la ligne horizontale a-b). Voici le pavage avant glissement :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a | b | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
... et après glissements successifs d’une case vers le bas des colonnes a-b :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Ce pavage n’est pas le même que le précédent ; on voit apparaître ici le nombre paveur ab7777777 soit 227777777. Lequel produit ce motif :
22777777722777777722777777722777777722777777722777777722777777722777777722777777
77722777777722777777722777777722777777722777777722777777722777777722777777722777
77777722777777722777777722777777722777777722777777722777777722777777722777777722
22777777722777777722777777722777777722777777722777777722777777722777777722777777
77722777777722777777722777777722777777722777777722777777722777777722777777722777
77777722777777722777777722777777722777777722777777722777777722777777722777777722
22777777722777777722777777722777777722777777722777777722777777722777777722777777
77722777777722777777722777777722777777722777777722777777722777777722777777722777
77777722777777722777777722777777722777777722777777722777777722777777722777777722
22777777722777777722777777722777777722777777722777777722777777722777777722777777
77722777777722777777722777777722777777722777777722777777722777777722777777722777
77777722777777722777777722777777722777777722777777722777777722777777722777777722
22777777722777777722777777722777777722777777722777777722777777722777777722777777
77722777777722777777722777777722777777722777777722777777722777777722777777722777
77777722777777722777777722777777722777777722777777722777777722777777722777777722
Examinons à présent le cas où les cases a et b se touchent par un sommet :
| | | | | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | × | × | × | |
--+---+---+---+---+---+---+--
| | × | a | × | × | |
--+---+---+---+---+---+---+--
| | × | × | b | × | |
--+---+---+---+---+---+---+--
| | × | × | × | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | | | | |
Cette disposition permet le pavage suivant :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Pas de nombre paveur en vue ; il faut donc faire glisser. Essayons l’horizontale pour le couple de rangées a/b (la verticale donnerait le même résultat), avec déplacement vers la droite :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Rebelote, le a jaune du bas est à la verticale de lui-même (le a souligné du haut). On remplace le couple a-b par le couple 2-2 et le nombre paveur 2777777272 apparaît :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 | 2 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
Voici le pavage que produit 2777777272 après redressement de 90° :
27777772722777777272277777727227777772722777777272277777727227777772722777777
77772722777777272277777727227777772722777777272277777727227777772722777777272
72722777777272277777727227777772722777777272277777727227777772722777777272277
22777777272277777727227777772722777777272277777727227777772722777777272277777
77777272277777727227777772722777777272277777727227777772722777777272277777727
77272277777727227777772722777777272277777727227777772722777777272277777727227
72277777727227777772722777777272277777727227777772722777777272277777727227777
77777727227777772722777777272277777727227777772722777777272277777727227777772
77727227777772722777777272277777727227777772722777777272277777727227777772722
27227777772722777777272277777727227777772722777777272277777727227777772722777
27777772722777777272277777727227777772722777777272277777727227777772722777777
77772722777777272277777727227777772722777777272277777727227777772722777777272
72722777777272277777727227777772722777777272277777727227777772722777777272277
22777777272277777727227777772722777777272277777727227777772722777777272277777
77777272277777727227777772722777777272277777727227777772722777777272277777727
77272277777727227777772722777777272277777727227777772722777777272277777727227
72277777727227777772722777777272277777727227777772722777777272277777727227777
77777727227777772722777777272277777727227777772722777777272277777727227777772
77727227777772722777777272277777727227777772722777777272277777727227777772722
27227777772722777777272277777727227777772722777777272277777727227777772722777
Si l’on avait déplacé les couples de lignes a\b vers la gauche, l’on aurait obtenu ceci :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a | 7 | 7 | a |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
... faisant apparaître (verticalement à droite – après remplacement des lettres par le chiffre 2), le nombre paveur 277727777, lequel produit le motif suivant, après rotation de 90° :
2777277772777277772777277772777277772777277772777277772777277772777277772777
7772777277772777277772777277772777277772777277772777277772777277772777277772
7277772777277772777277772777277772777277772777277772777277772777277772777277
2777277772777277772777277772777277772777277772777277772777277772777277772777
2777277772777277772777277772777277772777277772777277772777277772777277772777
7772777277772777277772777277772777277772777277772777277772777277772777277772
7277772777277772777277772777277772777277772777277772777277772777277772777277
2777277772777277772777277772777277772777277772777277772777277772777277772777
2777277772777277772777277772777277772777277772777277772777277772777277772777
7772777277772777277772777277772777277772777277772777277772777277772777277772
7277772777277772777277772777277772777277772777277772777277772777277772777277
2777277772777277772777277772777277772777277772777277772777277772777277772777
2777277772777277772777277772777277772777277772777277772777277772777277772777
7772777277772777277772777277772777277772777277772777277772777277772777277772
7277772777277772777277772777277772777277772777277772777277772777277772777277
2777277772777277772777277772777277772777277772777277772777277772777277772777
Lars Blomberg a remarqué que la position :
| | | | | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | × | × | × | |
--+---+---+---+---+---+---+--
| | × | a | × | × | |
--+---+---+---+---+---+---+--
| | × | × | b | × | |
--+---+---+---+---+---+---+--
| | × | × | × | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | | | | |
... permet de paver le plan ainsi :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 | 7 | 7 | a | 7 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b | 7 | 7 | 7 | b |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
... faisant apparaître le nombre paveur 3777 (une fois les a et les b, désormais liés en oblique infinie, remplacés par le chiffre 3).
Examinons enfin le cas où a et b ne se « touchent » pas. On voit ci-dessous que b ne peut occuper qu’une des trois positions r, s ou t – sous peine d’être trop éloigné de a :
| | | | | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | a | | | |
--+---+---+---+---+---+---+--
| | | | | | |
--+---+---+---+---+---+---+--
| | | r | s | t | |
--+---+---+---+---+---+---+--
| | | | | | |
Les positions a-r et a-t ne donnent rien – qu’on répète horizontalement leurs motifs ou qu’on les décale suivant une oblique. La position a-s produit en revanche le pavage en « saut de cavalier » (en référence au jeu d’échecs) – et 17777 pave ce plan (tant horizontalement que verticalement) :
177771777717777177771777717777177771777717777177771777717777177771777717777177
777177771777717777177771777717777177771777717777177771777717777177771777717777
717777177771777717777177771777717777177771777717777177771777717777177771777717
777717777177771777717777177771777717777177771777717777177771777717777177771777
771777717777177771777717777177771777717777177771777717777177771777717777177771
177771777717777177771777717777177771777717777177771777717777177771777717777177
777177771777717777177771777717777177771777717777177771777717777177771777717777
717777177771777717777177771777717777177771777717777177771777717777177771777717
777717777177771777717777177771777717777177771777717777177771777717777177771777
771777717777177771777717777177771777717777177771777717777177771777717777177771
177771777717777177771777717777177771777717777177771777717777177771777717777177
777177771777717777177771777717777177771777717777177771777717777177771777717777
717777177771777717777177771777717777177771777717777177771777717777177771777717
777717777177771777717777177771777717777177771777717777177771777717777177771777
771777717777177771777717777177771777717777177771777717777177771777717777177771
177771777717777177771777717777177771777717777177771777717777177771777717777177
777177771777717777177771777717777177771777717777177771777717777177771777717777
717777177771777717777177771777717777177771777717777177771777717777177771777717
777717777177771777717777177771777717777177771777717777177771777717777177771777
771777717777177771777717777177771777717777177771777717777177771777717777177771
Quels sont les nombres paveurs qui contiennent le chiffre 8 ? Il faut examiner là aussi le voisinage de a, seule case libre d’un plan peuplé de 8 :
| | | | | | | |
--+---+---+---+---+---+---+---+--
| | | | | | | |
--+---+---+---+---+---+---+---+--
| | × | × | × | × | × | |
--+---+---+---+---+---+---+---+--
| | × | × | × | × | × | |
--+---+---+---+---+---+---+---+--
| | × | × | a | × | × | |
--+---+---+---+---+---+---+---+--
| | × | × | × | × | × | |
--+---+---+---+---+---+---+---+--
| | × | × | × | × | × | |
--+---+---+---+---+---+---+---+--
| | | | r | s | t | u |
--+---+---+---+---+---+---+---+--
| | | | | | | |
Placer un autre a en r, s, t, ou u revient au même, si l’on cherche un nombre « avec des 8 » qui pave le plan. On obtient dans tous les cas ce pavage-ci :
| | | | | | | | | | | | | | | | | | | | | |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 | 8 | 8 | 1 |
--+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+--
| | | | | | | | | | | | | | | | | | | | | |
... ce qui fait apparaître (verticalement à droite), le nombre paveur 188888888.
Lars Blomberg a trouvé tous les nombres inférieurs à 10^12 qui pavent le plan :
n Count of solutions below 10^n
1 1
2 2
3 6
4 11
5 32
6 66
7 109
8 194
9 2475
10 7236
11 11098
12 14880
Voici les 2475 nombres inférieurs un milliard qui pavent le plan – c’est donc le début de la suite P que nous cherchions :
P = 9,
99, 366, 636, 663, 999, 3777, 7377, 7737, 7773, 9999, 17777, 33555, 34554,
35355, 35535, 35553, 43455, 45543, 53355, 53535, 53553, 54345, 55335, 55353,
55434, 55533, 71777, 77177, 77717, 77771, 99999, 132323, 153535, 231323,
232313, 243434, 313232, 323132, 323231, 336666, 342434, 343424, 351535, 353515,
354545, 366366, 366663, 424343, 434243, 434342, 453545, 454535, 515353, 535153,
535351, 535454, 545354, 545453, 633666, 636636, 663366, 663663, 666336, 666633,
999999, 1255255, 1525255, 1535535, 1552525, 1552552, 1553355, 2155255, 2515525,
2525155, 2525515, 2551255, 2551525, 2552155, 2552551, 3355155, 3515355,
3551553, 3553515, 5125525, 5152525, 5153553, 5155252, 5155335, 5215525,
5251552, 5252515, 5252551, 5255125, 5255152, 5255215, 5335515, 5351535, 5355351,
5512552, 5515252, 5515533, 5521552, 5525251, 5525512, 5525521, 5533551,
5535153, 9999999, 13334444, 13344443, 13434434, 13444433, 14334344, 14343434,
14344343, 14434334, 14444333, 25454545, 31334444, 31344443, 31434434, 31444433,
33134444, 33144443, 33314444, 33344441, 33414434, 33434414, 33444413, 33444431,
34134344, 34143434, 34144343, 34314344, 34334144, 34341434, 34343414, 34344143,
34344341, 34414334, 34434134, 34434314, 34444133, 34444313, 34444331, 37773777,
41333444, 41343443, 41433434, 41434343, 41443433, 43133444, 43143443, 43313444,
43331444, 43341443, 43343441, 43413434, 43414343, 43431434, 43433414, 43434143,
43434341, 43441433, 43443413, 43443431, 44133344, 44143343, 44313344, 44331344,
44333144, 44341343, 44343143, 44343341, 44413334, 44431334, 44433134, 44433314,
44441333, 44443133, 44443313, 44443331, 45254545, 45452545, 45454525, 52545454,
54525454, 54545254, 54545452, 73777377, 77377737, 77737773, 99999999,
122666666, 126266666, 126626666, 126662666, 126666266, 126666626, 126666662,
133355555, 133535555, 133553555, 133555355, 133555535, 133555553, 135335555,
135353555, 135355355, 135355535, 135355553, 135533555, 135535355, 135535535,
135535553, 135553355, 135553535, 135553553, 135555335, 135555353, 135555533,
153335555, 153353555, 153355355, 153355535, 153355553, 153533555, 153535355,
153535535, 153535553, 153553355, 153553535, 153553553, 153555335, 153555353,
153555533, 155333555, 155335355, 155335535, 155335553, 155353355, 155353535,
155353553, 155355335, 155355353, 155355533, 155533355, 155533535, 155533553,
155535335, 155535353, 155535533, 155553335, 155553353, 155553533, 155555333,
162266666, 162626666, 162662666, 162666266, 162666626, 162666662, 166226666,
166262666, 166266266, 166266626, 166266662, 166622666, 166626266, 166626626, 166626662,
166662266, 166662626, 166662662, 166666226, 166666262, 166666622, 188888888,
212666666, 216266666, 216626666, 216662666, 216666266, 216666626, 216666662,
221666666, 223334444, 223343444, 223344344, 223344434, 223344443, 223433444,
223434344, 223434434, 223434443, 223443344, 223443434, 223443443, 223444334,
223444343, 223444433, 224333444, 224334344, 224334434, 224334443, 224343344,
224343434, 224343443, 224344334, 224344343, 224344433, 224433344, 224433434,
224433443, 224434334, 224434343, 224434433, 224443334, 224443343, 224443433,
224444333, 226166666, 226616666, 226661666, 226666166, 226666616, 226666661,
227777777, 232334444, 232343444, 232344344, 232344434, 232344443, 232433444,
232434344, 232434434, 232434443, 232443344, 232443434, 232443443, 232444334,
232444343, 232444433, 233234444, 233243444, 233244344, 233244434, 233244443,
233324444, 233342444, 233344244, 233344424, 233344442, 233423444, 233424344,
233424434, 233424443, 233432444, 233434244, 233434424, 233434442, 233442344,
233442434, 233442443, 233443244, 233443424, 233443442, 233444234, 233444243,
233444324, 233444342, 233444423, 233444432, 234233444, 234234344, 234234434,
234234443, 234243344, 234243434, 234243443, 234244334, 234244343, 234244433,
234323444, 234324344, 234324434, 234324443, 234332444, 234334244, 234334424,
234334442, 234342344, 234342434, 234342443, 234343244, 234343424, 234343442,
234344234, 234344243, 234344324, 234344342, 234344423, 234344432, 234423344,
234423434, 234423443, 234424334, 234424343, 234424433, 234432344, 234432434,
234432443, 234433244, 234433424, 234433442, 234434234, 234434243, 234434324,
234434342, 234434423, 234434432, 234442334, 234442343, 234442433, 234443234,
234443243, 234443324, 234443342, 234443423, 234443432, 234444233, 234444323,
234444332, 242333444, 242334344, 242334434, 242334443, 242343344, 242343434,
242343443, 242344334, 242344343, 242344433, 242433344, 242433434, 242433443,
242434334, 242434343, 242434433, 242443334, 242443343, 242443433, 242444333,
243233444, 243234344, 243234434, 243234443, 243243344, 243243434, 243243443,
243244334, 243244343, 243244433, 243323444, 243324344, 243324434, 243324443,
243332444, 243334244, 243334424, 243334442, 243342344, 243342434, 243342443,
243343244, 243343424, 243343442, 243344234, 243344243, 243344324, 243344342,
243344423, 243344432, 243423344, 243423434, 243423443, 243424334, 243424343,
243424433, 243432344, 243432434, 243432443, 243433244, 243433424, 243433442,
243434234, 243434243, 243434324, 243434342, 243434423, 243434432, 243442334,
243442343, 243442433, 243443234, 243443243, 243443324, 243443342, 243443423,
243443432, 243444233, 243444323, 243444332, 244233344, 244233434, 244233443,
244234334, 244234343, 244234433, 244243334, 244243343, 244243433, 244244333,
244323344, 244323434, 244323443, 244324334, 244324343, 244324433, 244332344,
244332434, 244332443, 244333244, 244333424, 244333442, 244334234, 244334243,
244334324, 244334342, 244334423, 244334432, 244342334, 244342343, 244342433,
244343234, 244343243, 244343324, 244343342, 244343423, 244343432, 244344233,
244344323, 244344332, 244423334, 244423343, 244423433, 244424333, 244432334,
244432343, 244432433, 244433234, 244433243, 244433324, 244433342, 244433423,
244433432, 244434233, 244434323, 244434332, 244442333, 244443233, 244443323,
244443332, 261266666, 261626666, 261662666, 261666266, 261666626, 261666662,
262166666, 262616666, 262661666, 262666166, 262666616, 262666661, 266126666,
266162666, 266166266, 266166626, 266166662, 266216666, 266261666, 266266166,
266266616, 266266661, 266612666, 266616266, 266616626, 266616662, 266621666,
266626166, 266626616, 266626661, 266661266, 266661626, 266661662, 266662166,
266662616, 266662661, 266666126, 266666162, 266666216, 266666261, 266666612,
266666621, 272777777, 277277777, 277727777, 277772777, 277777277, 277777727,
277777772, 313355555, 313535555, 313553555, 313555355, 313555535, 313555553,
315335555, 315353555, 315355355, 315355535, 315355553, 315533555, 315535355,
315535535, 315535553, 315553355, 315553535, 315553553, 315555335, 315555353,
315555533, 322334444, 322343444, 322344344, 322344434, 322344443, 322433444,
322434344, 322434434, 322434443, 322443344, 322443434, 322443443, 322444334,
322444343, 322444433, 323234444, 323243444, 323244344, 323244434, 323244443,
323324444, 323342444, 323344244, 323344424, 323344442, 323423444, 323424344,
323424434, 323424443, 323432444, 323434244, 323434424, 323434442, 323442344,
323442434, 323442443, 323443244, 323443424, 323443442, 323444234, 323444243,
323444324, 323444342, 323444423, 323444432, 324233444, 324234344, 324234434,
324234443, 324243344, 324243434, 324243443, 324244334, 324244343, 324244433,
324323444, 324324344, 324324434, 324324443, 324332444, 324334244, 324334424,
324334442, 324342344, 324342434, 324342443, 324343244, 324343424, 324343442,
324344234, 324344243, 324344324, 324344342, 324344423, 324344432, 324423344,
324423434, 324423443, 324424334, 324424343, 324424433, 324432344, 324432434,
324432443, 324433244, 324433424, 324433442, 324434234, 324434243, 324434324,
324434342, 324434423, 324434432, 324442334, 324442343, 324442433, 324443234,
324443243, 324443324, 324443342, 324443423, 324443432, 324444233, 324444323,
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661666622, 662126666, 662162666, 662166266, 662166626, 662166662, 662216666, 662261666,
662266166, 662266616, 662266661, 662612666, 662616266, 662616626, 662616662,
662621666, 662626166, 662626616, 662626661, 662661266, 662661626, 662661662,
662662166, 662662616, 662662661, 662666126, 662666162, 662666216, 662666261,
662666612, 662666621, 663336666, 663363666, 663366366, 663366636, 663366663,
663633666, 663636366, 663636636, 663636663, 663663366, 663663636, 663663663,
663666336, 663666363, 663666633, 666122666, 666126266, 666126626, 666126662,
666162266, 666162626, 666162662, 666166226, 666166262, 666166622, 666212666,
666216266, 666216626, 666216662, 666221666, 666226166, 666226616, 666226661,
666261266, 666261626, 666261662, 666262166, 666262616, 666262661, 666266126,
666266162, 666266216, 666266261, 666266612, 666266621, 666333666, 666336366,
666336636, 666336663, 666363366, 666363636, 666363663, 666366336, 666366363,
666366633, 666612266, 666612626, 666612662, 666616226, 666616262, 666616622,
666621266, 666621626, 666621662, 666622166, 666622616, 666622661, 666626126,
666626162, 666626216, 666626261, 666626612, 666626621, 666633366, 666633636,
666633663, 666636336, 666636363, 666636633, 666661226, 666661262, 666661622,
666662126, 666662162, 666662216, 666662261, 666662612, 666662621, 666663336,
666663363, 666663633, 666666122, 666666212, 666666221, 666666333, 722777777,
727277777, 727727777, 727772777, 727777277, 727777727, 727777772, 772277777,
772727777, 772772777, 772777277, 772777727, 772777772, 777227777, 777272777,
777277277, 777277727, 777277772, 777722777, 777727277, 777727727, 777727772,
777772277, 777772727, 777772772, 777777227, 777777272, 777777722, 818888888,
881888888, 888188888, 888818888, 888881888, 888888188, 888888818, 888888881,
999999999,...
À noter que ces 2475 nombres ne produisent que 282 pavages aux motifs distincts – illustrés ici :
9, 366, 3777, 17777, 33555, 34554, 35355, 132323, 153535, 243434, 336666, 354545, 1255255, 1525255, 1535535, 1552525, 1552552, 1553355, 9999999, 13334444, 13344443, 13434434, 13444433, 14334344, 14343434, 14344343, 14434334, 14444333, 25454545, 122666666, 126266666, 126626666, 126662666, 126666266, 126666626, 126666662, 133355555, 133535555, 133553555, 133555355, 133555535, 133555553, 135335555, 135353555, 135355355, 135355535, 135355553, 135533555, 135535355, 135535535, 135535553, 135553355, 135553535, 135553553, 135555335, 135555353, 135555533, 153335555, 153353555, 153355355, 153355535, 153355553, 153533555, 153535355, 153535535, 153535553, 153553355, 153553535, 153553553, 153555335, 153555353, 153555533, 155333555, 155335355, 155335535, 155335553, 155353355, 155353535, 155353553, 155355335, 155355353, 155355533, 155533355, 155533535, 155533553, 155535335, 155535353, 155535533, 155553335, 155553353, 155553533, 155555333, 162266666, 162626666, 162662666, 162666266, 162666626, 162666662, 166226666, 166262666, 166266266, 166266626, 166266662, 166622666, 166626266, 166626626, 166626662, 166662266, 166662626, 166662662, 166666226, 166666262, 166666622, 188888888, 223334444, 223343444, 223344344, 223344434, 223344443, 223433444, 223434344, 223434434, 223434443, 223443344, 223443434, 223443443, 223444334, 223444343, 223444433, 224333444, 224334344, 224334434, 224334443, 224343344, 224343434, 224343443, 224344334, 224344343, 224344433, 224433344, 224433434, 224433443, 224434334, 224434343, 224434433, 224443334, 224443343, 224443433, 224444333, 227777777, 232334444, 232343444, 232344344, 232344434, 232344443, 232433444, 232434344, 232434434, 232434443, 232443344, 232443434, 232443443, 232444334, 232444343, 232444433, 233234444, 233243444, 233244344, 233244434, 233244443, 233324444, 233342444, 233344244, 233344424, 233423444, 233424344, 233424434, 233424443, 233432444, 233434244, 233434424, 233442344, 233442434, 233442443, 233443244, 233443424, 233444234, 233444243, 233444324, 234234344, 234234434, 234234443, 234243344, 234243434, 234243443, 234244334, 234244343, 234244433, 234323444, 234324344, 234324434, 234324443, 234332444, 234334244, 234334424, 234342344, 234342434, 234342443, 234343244, 234343424, 234344243, 234344324, 234423443, 234424334, 234424343, 234424433, 234432434, 234432443, 234433244, 234433424, 234434243, 234434324, 234442433, 234443243, 234443324, 242433344, 242433434, 242433443, 242434334, 242434343, 242434433, 242443334, 242443343, 242443433, 242444333, 243243344, 243243434, 243243443, 243244334, 243244343, 243244433, 243324344, 243324434, 243324443, 243332444, 243334244, 243342434, 243342443, 243343244, 243424343, 243424433, 243432443, 243433244, 244244333, 244324433, 272777777, 277277777, 277727777, 333666666, 336366666, 336636666, 336663666, 336666366, 336666636, 353554455, 363636666, 363663666, 363666366, 444455555, 444545555, 444554555, 444555455, 444555545, 445445555, 445454555, 445455455, 445455545, 445544555, 445545455, 445545545, 445554545, 454545455.
Merci encore à Lars Blomberg et Olivier Miakinen pour leurs calculs et travaux – et rappelons aux amateurs que le dénombrement des « pavages » des boîtes dans R3 (leur remplissage, plutôt), reste une question ouverte, à la combinatoire explosive au delà du cas 2 x 2 x 2 !
à+
É.
Bruxelles, le 29 novembre 2011.
__________
[Éric, sur fsm] :
> Je t’avoue que je visionne mal dans l’espace...
[Olivier Miakinen] :
Alors revenons temporairement au plan. Je choisis l’un de tes résultats au hasard, par exemple que 132323 pave le plan :
========================================================================
1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3
2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3
2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3
1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3
2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3
2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3
1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3
2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3
2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3 2 3 1 3 2 3
========================================================================
Multiplie tous les entiers par 2 :
========================================================================
2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6
4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6
4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6
2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6
4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6
4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6
2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6
4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6
4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6 4 6 2 6 4 6
========================================================================
Maintenant colle une deuxième couche identique sur la première, et tu as un pavage infini d’une « peau » d’épaisseur 2, infinie dans les deux autres directions. C’est-à-dire un pavage de { 0, 1 } x IR x IR.
Je ne l’ai pas vérifié, mais il doit être possible d’appliquer ta technique du glissement sur le deuxième plan (ou même sur des lignes individuelles du deuxième plan).
Et si au lieu de multiplier les nombres par 2 tu les multiplies par 3 :
========================================================================
3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9
6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9
6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9
3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9
6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9
6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9
3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9
6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9
6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9 6 9 3 9 6 9
========================================================================
Tu colles une infinité de plans identiques, et tu obtiens un pavage infini de IR³.
Là encore, la technique du glissement devrait permettre d’éviter les répétitions infinies d’un seul chiffre, dans toutes les directions.
Et bien entendu, cette technique (x2 et x3) fonctionne avec l’ensemble de tes résultats dans le plan, aussi bien les rectangles et les rubans infinis que les remplissages complets du plan IR².
------------------------------------------------------------------------
Maintenant, à toi de bosser. Trouve-moi un remplissage avec des 5 dans un rectangle de dimensions finies (ce qui exclut donc tes rubans infinis, horizontaux ou obliques). Si tu y arrives, colles-en deux couches l’une sur l’autre, et tu auras la preuve que l’on peut mettre un 10 dans une boîte finie. À partir de ce résultat, s’il existe, il ne devrait pas être si difficile de retirer quelques nombres pour en faire des 9.
Merci, Olivier ! Je cherche...
__________
Dernière mise à jour :
21 décembre 2011.
(pour
nous écrire, c’est toujours eric pavage angelini chez
skynet pavage be – avec
point pour pavage)