# Contiguous Partridge Tilings

This formula holds for every number n:

Σ(i=1; n) i3 = T(n)2,

where T(n) is the nth triangular number. This formula implies that a square with side T(n) might be dissected into 1 square of side 1, 2 squares of side 2, and so on up to n squares of side n. Such a tiling is a partridge tiling, after the partridge in the song The Twelve Days of Christmas.

Robert Wainwright was the first to propose partridge tilings, and the first to find one: a square of side T(12), dissected in partridge fashion into squares with sides 1 through 12. Partridge tilings were later extended to shapes other than squares. See for example the August 2002 issue of Erich Friedman's Math Magic.

In 2020 I found this minimal partridge tiling of the monodom:

This is a contiguous partridge tiling. The tiles of each size form a connected group. This is the only contiguous partridge tiling of the monodom with 6 sizes (the minimum), except that any rectangle formed by two equal pieces may be flipped.

Joyce Michel has pointed out that the groups are connected in series: the 1 is adjacent to the 2's, the 2's to the 3's, and so on in a clockwise spiral.

Joyce also extended the above tiling to use 7 sizes:

This tiling also has the groups connected in series in a clockwise spiral.

The only other contiguous partridge tiling that I know of is Michael Reid's trapezoidal tiling with 4 sizes:

I have found no other contiguous partridge tiling of any plane figure, whether or not the tiling is minimal. If you find one, please write.