We can also look for Border Sub-Partridge Tilings, a tiling of a rectangle with squares with a border of width 1 around each square. For example, a {1,3,4,5,7} border tiling is shown below. What border sub-partridge tilings exist?

The second is {2,4,6,...2n}, as shown by the following tiling:

The third is {4,8,24,60,...4F_nF_n+1} (where F_n is the n^th Fibonacci number), as shown by the following tiling:

Jeremy Galvagni showed that {a²bc,ab²c} tilings exist. He conjectured that these are the only sub-partridge tilings using only two sizes of squares.

Luke Pebody showed that this conjecture isn't true by finding the {35,70} tiling below. He and Emilio Schiavi showed that an {n,2n} tiling does not exist precisely when n is an odd prime or n=1, 9, 15, 21, 25, 27, or 33.

Patrick Hamlyn found many sub-partridge tilings with his tiling program.

Patrick Hamlyn and Jeremy Galvagni noted that border tilings are equivalent to tiling n-1 copies of each square of side n without borders.

Claudio Baiocchi notes that if there is a {a₁, a₂, . . . a_n} tiling, then there is a {k a₁, k a₂, . . . k a_n} tiling for every k, and that for every {a₁, a₂, . . . a_n}, there exists some k so that there is a {k a₁, k a₂, . . . k a_n} tiling.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/31/06.