n \ m | 0 | 1 | 2 | 3 |
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0 | 0 / 0 | |||
1 | φ | φ | ||
2 | 1 / 1 | 4 / 8 | 6 / 18 | |
3 | φ | φ | 9 / 31 | 17 / 46 (BH/AS) |
4 | 2 / 2 | φ | 5 / 44 (EF/DB) | ? |
5 | φ | 14 / 20 | 8 / 47 | ? |
6 | 3 / 3 | φ | 10 / 57 | ? |
7 | φ | 16 / 34 | 13 / 70 | ? |
8 | 4 / 4 | 21 / 27 | 15 / 87 | ? |
9 | φ | 20 / 48 | 18 / 90 | ? |
Anti Solg and Bruce Herdt sent many 3×3 grids.
Luke Pebody showed that the smallest totals for large 1×n grids depend on n mod 3, and that the largest totals for large 1×n grids depend on n mod 2:
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Richard Sabey gave this small total tiling:
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Luke Pebody found this group of 7's that tiles the plane, giving him hope that for large m×n grids, a total on the order of 7(2mn+m+n)/3 is possible:
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David Bevan found the largest totals for 2n×4 and 4n×6 grids:
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The 0×n solutions are trivial. Here are the known 1×n solutions for small n:
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Luke Pebody gave a full analysis of the 1×n possibilities, essentially agreeing with the following directed graph of mine:
Here are the known 2×n solutions for small n:
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If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/12/08.