# Lovebirds Tilings

## Introduction

This formula holds for every number *k*:

Σ(*i*=1; *k*) *i*^{3}
= *T*(*k*)^{2},
where *T*(*k*) is the *k*th triangular number.
This formula implies that a square with side *T*(*k*)
might be dissected into 1 square of side 1, 2 squares of side 2,
and so on up to *k* squares of side *k*.
Such a tiling is a *partridge tiling*,
named after the partridge in the song The Twelve Days of Christmas.

It was first proposed and discovered by Robert Wainwright.
See the August 2002
issue
of Erich Friedman's *Math Magic*
for a survey of known partridge tilings.

In 2021 I introduced *lovebirds tilings.*
These use a *double* set of copies of a shape,
with scale factors 1…*k,* to construct
two copies of the shape at scale *T*(*k*).

Any plane shape with a partridge tiling can use two copies
of the tiling to form a lovebirds tiling.
Here I show polyforms that can form lovebirds tilings
but not partridge tilings,
or can form lovebirds tilings with fewer sizes of tiles than their smallest
known partridge tilings.
If you find a new lovebirds tiling with this property, please write.

## Polyominoes

The monomino, or square, has a minimal partridge tiling with *k*=8.
It has a minimal lovebirds tiling with *k*=7:

The domino has partridge number 7.
It has lovebirds number 3:

It follows that any parallelogram with sides in the ratio 1:2
has a lovebirds number of 3 or less.
This includes the straight tetriamond.

The I tetromino has partridge number 7.
Its lovebirds number is 6:

The L tetromino has no known partridge tiling.
Its lovebirds number is 3:

The P hexomino has no known partridge tiling.
Its lovebirds number is 3:

The 2×3 rectangle has partridge number 7.
It has lovebirds number 3:

The 3×4 rectangle has partridge number 7.
It has lovebirds number 3:

## Polyiamonds

The moniamond has partridge number 9.
Its lovebirds number is 7:

It follows that any triangle has lovebirds number 7 or less.

## Polyaboloes

The monabolo has partridge number 8.
Erich Friedman discovered that it has lovebirds number 3:

From this Erich deduced that the lovebirds number
of any right triangle is at most 3.

## Polydoms

The monodom has partridge number 6.
Its lovebirds number is 3:

This tridom has partridge number 6.
Its lovebirds number is 4:

## Polydrafters

The monodrafter has partridge number 4.
Its lovebirds number is 3:

This tridrafter has partridge number 6 and lovebirds number 3:

*Last revised 2021-04-07.*

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Polyform Curiosities

Col. George Sicherman
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