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### Brickyard

This page presents statistics on solutions with six diamonds and six squares arrayed as they might be in brick wall.

For this pattern, the solid-color border between the diamonds and the squares is two L-shaped lines.

For different placements of the solids, the numbers vary greatly, from under 100 to well over 10000. The total number of solutions is 119,469.

The largest numbers of solutions (roughly half the total) occur when solids appear in positions 8 and 15, as diamonds in the center of the board, each bordering two squares.

The smallest numbers of solutions occur when one or more solids sit as squares bordering the diamonds (10,11, 12,13).

As shown below, solids NEVER occupy 10 of the 24 positions (5 symmetric pairs).

In the table below, the solid placements in the third and fourth columns have been translated from what is shown in the second part of the applet above in order to have all positions relative to the same orientation of the pattern.

The cases with the most and fewest solutions are highlighted.
 In the figure at the left, the positions where the solids never appear are shaded. Positions marked with * are those with solids only on squares. Only (0,8,15) has solids only on diamonds. ``` 0,3,10 385 0,3,11 124 0,3,15 1019 0,3,18 334 0,3,19 448 0,3,23 201 0,4,8 3347 0,4,11 234 0,4,12 359 0,4,13 147 0,4,15 1392 0,4,18 503 0,4,19 480 0,4,20 186 0,4,23 246 0,5,8 2704 0,5,12 568 0,5,13 221 0,5,18 652 0,5,19 462 0,5,20 130 ``` ``` 0,8,10 1196 0,8,11 3381 0,8,15 21632 0,8,18 2189 0,8,19 2469 0,8,20 2965 0,10,12 31 0,10,18 124 0,10,19 21 0,10,23 94 0,11,12 258 0,11,18 299 0,11,19 95 0,11,23 330 0,12,15 2497 0,12,19 231 0,12,20 559 0,13,15 987 0,13,20 1051 0,15,19 1049 ``` ``` 0,13,18 298 0,15,18 2025 0,18,20 102 0,18,23 304 3,10,18 848* 3,10,19 923* 3,11,18 193* 3,11,19 228* 3,15,18 1770 3,15,19 2352 3,18,20 56* 4,11,18 564* 4,12,19 470* 4,13,18 122* 4,15,18 2514 4,15,19 2373 4,18,20 308* 4,8,18 2265 ``` ``` 5,13,18 225* 5,15,18 2472 5,18,20 228* 8,10,18 653 8,10,19 521 8,11,18 2052 8,11,19 1755 8,15,18 14819 8,15,19 15536 8,18,20 1550 8,18,23 1223 10,12,19 14* 11,12,19 195* 12,15,19 2563 13,15,18 1293 13,18,20 830* 13,18,23 41* 18,20,23 314 ```