Cutting Two Squares to Make One Square

1. General Squares

Problem: Design an algorithm that takes two squares (of arbitrary relative size) and cuts them into a total of 5 pieces in such a way that the resulting pieces can be rearranged to form a single square.

A perl script that takes the sizes of the two squares as arguments and generates a 1-page PostScript file containing two different solutions:
- The first solution cuts the larger square into 4 pieces and leaves the smaller square uncut. The pieces are only translated (never rotated) in forming the new square.
- The second solution cuts the larger square into 3 pieces and the smaller square into 2 pieces. The pieces are only rotated (never translated) in forming the new square.
The script draws the two squares in thick lines, the cuts in thin lines and the rearrangement in dashed lines.
A sample PostScript output that shows the solutions for 9 pairs of squares with relative sizes ranging from 8:0 to 8:8 in steps of 8:1. (This file is the result of joining together 9 separate files generated by the perl script. The page numbers indicate the relative sizes of the two squares.)

2. Pythagorean Squares

Problem: The cutting schemes given above work for all pairs of squares, but require 5 pieces. Are there classes of squares for which the problem can be solved in fewer than 5 pieces?

A triple [a, b, c] where a, b and c are positive integers with a² + b² = c² is called a Pythagorean triple for obvious reasons. If, in addition, a and b are relatively prime, [a, b, c] is called a fundamental Pythagorean triple (FPT). Thus, among the Pythagorean triples [8, 6, 10], [15, 8, 17] and [50, 120, 130], only [15, 8, 17] is an FPT. All [a, b, c] forming FPTs are generated by the parametric equations

a = 2mn
b = m² - n²
c = m² + n²

where m and n are any relatively prime positive integers, not both odd, and m > n. A proper subset of FPTs can be obtained by substituting m = n + 1

a = 2n² + 2n
b = 2n + 1
c = 2n² + 2n + 1

where n is any positive integer (the substitution took care of all constraints on m and n).

The Java applet below shows animated 4-piece solutions for squares that have sides in the ratio a/b for some a and b in this subset of FPTs.

Though the applet is written to work for all values of n, the available screen real-estate puts an upper bound on what can be displayed. The largest usable value of n is computed to be the one that will make each unit cell two pixels wide (with just one colored pixel between every pair of adjacent white grid lines).

Usage Guide

Use the "+" and "-" buttons to select the sizes of the squares.
Then use the action button (the rightmost one) to cycle through the steps of drawing the squares, cutting them and reassembling the pieces.
The action button can be used to preempt the animation: if it is pressed while the animation is in progress, the animating thread is killed and display area is cleared.
The "+" and "-" buttons preempt the state of the action button. When "+" or "-" is pressed to change the size, the drawing area is cleared and the action button resets to the "Draw Squares" state.

I solved the problem first for [4, 3, 5] and [12, 5, 13] and then generalized the solution to the form used in the applet. I don't know whether 4-piece solutions are possible for FPTs such as [15, 8, 17] and [20, 21, 29] that are not covered by this scheme.

See Elementary Number Theory by Underwood Dudley for the derivation of the first set of parametric equations (in terms of m and n) given above for FPTs.

V. B. Balayoghan (vbb@cs.utexas.edu)