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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