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### 3x8 Solutions

##### Square Stairs or Diamond Mountain
This page shows a 3x8 rectangle with complete results for two rotations of a pattern called "Square Stairs" or "Diamond Mountain".

The total numbers of solutions for them are 34103 and 54116. Of those, 4344 were found for each pattern, because the solids were placed symmetrically with regard to the middle row (either opposite each other above and below, with the third in the middle row, or all three in the middle row). Adjusting for those, the totals are 29759 + 54116 = 83875.

The numbers of solutions vary greatly for different placements of the solids, from over 5000 down to a mere pair.
Finding those few by hand would be a real challenge.

A table with all the statistics is at the bottom of this page.

##### Double Bars
It is interesting to note that making the smallest of changes to the pattern, namely swapping one diamond and one square on each side to make four columns of squares and four columns of diamonds, makes an enormous difference in the numbers of solutions.

With the latter pattern, just the first two solid positions, 0,2,4 and 0,2,5 each exceed the combined total above, with 105922 and 428851 solutions respectively. Moving the third solid just one space, so that it faces a square placement, makes a huge difference just between those two!

The solutions presented here are a tiny! sample of the thousands of solutions. Some are cylinders (right and left edges match), some have high symmetry. Some have "rings" and "rabbits".

 ```1,15,20 5103 S 1,3,22 4660 Mt. 1,8,20 4121 S 1,4,17 3894 Mt. 1,8,13 3340 S 1,6,8 3154 S 0,17,22 2587 Mt. 1,10,22 2019 Mt. 1,6,17 1940 S 1,13,17 1752 BOTH 1,3,5 1736 Mt. 2,4,15 1705 Mt. 1,13,15 1629 S 0,4,17 1551 Mt. 1,15,21 1515 S 3,8,14 1484 Mt. 1,4,20 1468 S 0,10,22 1458 Mt. 1,3,14 1428 Mt. 2,8,13 1410 Mt. 0,3,22 1401 Mt. 1,3,13 1244 Mt. 2,4,20 1233 Mt. 1,8,14 1194 S 0,13,17 1192 Mt. 3,14,19 1056 BOTH 2,4,8 1049 Mt. 1,8,21 988 S 0,2,22 977 Mt. 0,14,17 968 Mt. 0,5,10 947 Mt. 1,5,10 935 Mt. 0,10,14 809 Mt. ``` ```3,9,20 902 S 0,5,17 876 Mt. 0,9,22 816 Mt. 1,10,14 722 Mt. 2,4,19 686 Mt. 0,9,13 642 Mt. 1,3,20 616 S 0,2,13 611 Mt. 0,3,5 592 Mt. 0,3,14 571 Mt. 1,14,17 568 BOTH 0,9,14 564 Mt. 1,4,10 562 Mt. 0,4,9 548 Mt. 2,13,15 507 Mt. 2,5,8 503 Mt. 1,4,14 500 S 0,2,4 469 Mt. 1,5,17 469 Mt. 0,2,14 439 Mt. 0,5,9 419 Mt. 2,8,14 410 Mt. 1,3,13 409 S 0,2,5 397 Mt. 8,10,14 376 BOTH 0,4,10 375 Mt. 3,8,13 370 Mt. 0,3,13 321 Mt. 3,13,15 310 Mt. 3,13,19 296 BOTH 1,3,6 288 S 1,3,19 277 Mt. 1,3,8 252 Mt. 1,4,21 238 S 0,6,17 232 S 0,17,20 229 S 3,9,13 216 S ``` ```0,15,20 209 S 2,13,19 202 Mt. 0,13,17 186 S 1,8,10 173 Mt. 0,20,22 167 S 1,8,22 154 S 1,6,10 144 Mt. 1,3,6 133 Mt. 1,13,15 132 Mt. 0,4,20 126 S 0,15,22 124 Mt. 8,10,15 118 BOTH 1,6,17 116 Mt. 3,8,19 111 BOTH 0,13,22 106 S 1,3,20 106 Mt. 0,6,23 105 S 0,17,21 87 S 1,5,15 82 Mt. 0,13,15 78 Mt. 1,8,15 76 S 0,14,17 72 S 0,6,13 68 Mt. 0,6,15 64 S 0,3,23 63 Mt. 0,6,22 62 Mt. 0,6,16 60 S 0,13,16 59 BOTH 0,13,15 56 S 0,10,23 55 Mt. 0,4,6 53 Mt. 1,10,15 50 Mt. 1,3,15 50 Mt. 0,5,15 47 Mt. 0,4,15 43 Mt. 0,3,20 42 S 1,4,15 42 Mt. 0,3,6 40 S ``` ```0,2,23 39 Mt. 0,15,21 38 S 1,4,8 38 S 3,8,20 38 Mt. 0,6,17 34 Mt. 1,10,20 34 Mt. 0,7,10 31 Mt. 0,7,17 30 Mt. 0,3,6 28 Mt. 0,3,7 28 Mt. 0,6,10 28 Mt. 2,8,15 28 Mt. 0,4,16 25 Mt. 0,15,17 24 Mt. 0,6,9 24 Mt. 3,8,15 24 Mt. 0,4,20 17 Mt. 1,6,8 16 Mt. 0,17,20 14 Mt. 0,7,9 14 Mt. 0,9,23 14 Mt. 0,6,22 13 S 1,8,15 12 Mt. 3,13,15 12 S 0,3,15 11 Mt. 0,5,16 11 Mt. 0,3,13 10 S 0,10,15 9 Mt. 0,14,16 8 BOTH 0,7,17 8 S 0,6,23 7 Mt. 0,17,22 6 S 0,13,19 5 Mt. 0,2,7 4 Mt. 0,3,19 3 Mt. 0,4,19 2 Mt. 0,7,16 2 S ```