where *T*(*n*) is the *n*th 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.

See also Contiguous Reverse Partridge Tilings.

*Last revised 2023-07-13.*

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Col. George Sicherman [ HOME | MAIL ]