# Contiguous Reverse Partridge Tilings

## Introduction

This formula holds for every number *n*:

Σ(*i*=1; *n*) *i*^{3}
= *T*(*n*)^{2},
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 called 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.

A *reverse partridge tiling*
is a dissection
in which the number of tiles decreases as the scale of the tile
increases.
For some values of *n* there is a number *k* such that

*k*^{2} = Σ(*i*=1; *n*)
*i*^{2}(*n*+1−*i*).
The first two values of *n* for which *k* is an integer
are 6 (*k*=14) and 25 (*k*=195).
Many polygons are known to have reverse partridge tilings for
*k*=14, and a few for *k*=195.
You can see them at Erich Friedman's
Math
Magic page for June 2007.
They include solutions for the square, the equilateral triangle (and
hence any triangle), the triamond, the L tetromino, and the monodrafter.

Here I show all known reverse partridge tilings with *n*=6
in which equal
tiles are contiguous.
According to Wainwright, this subproblem has apparently not been studied
before.
If you solve any cases not shown here, please
write.

## Polyominoes

The monomino, or square, has two contiguous solutions:

By transforming these solutions one may obtain
a solution for any rectangle or parallelogram.

## Polytans

The moniamond, or equilateral triangle,
has no contiguous solution with 6 scales.
If it did, such a solution could be distorted to solve other
triangles such as the monotan, the monodrafter, and the monodom.
The smallest solution for the monotan, or monabolo,
uses sizes 1 through 4 to make a monotan at scale 5√2.
This format has no contiguous solutions.
But with sizes 1 through 6 there are many solutions.
Here is one:

The right trapezoidal tritan also has many solutions.

The parallelogrammatic ditan has a solution with *n*=4:

## Polyiamonds

The triamond has many solutions:

## Polydrafters

The monodrafter has at least one solution:

## Polydoms

The monodom has several solutions.

This parallelogrammatic didom and tetradom can be solved with fewer scales
than the monomino:

*Last revised 2021-06-02.*

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Col. George Sicherman
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