n \ m  0  1  2  3  4  5 

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 (/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:

Richard Sabey gave this small total tiling:

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:

David Bevan found the largest totals for 2n×4 and 4n×6 grids:

The 0×n solutions are trivial. Here are the known 1×n solutions for small n:





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:


If you can extend any of these results, please email me. Click here to go back to Math Magic. Last updated 8/12/08.