Cutting Squares

Here is a problem:

For N = 1, the question is vacuous. For N = 2, we actually have several of solution. The keyword here is the Pythagorean Theorem. Solutions ##2, 3, 4, 14, 15, 26, 27, 28 all show different ways to cut two squares into pieces that combine into a single square.

What about larger N? The answer is yes, of course. To solve the problem for N>2 all we have to know is how to solve it for N = 2. The argument is inductive. Assume we know how to solve the problem for some N = K>2. Consider K+1 squares. Pick out any two of them and, by dissection, combine them into a bigger square. After this step, we are left with only K squares, and the inductive assumption implies.

Reference

G.N.Frederickson, Dissections: Plane & Fancy, Cambridge University Press, 1997

Copyright © 1996-2009 Alexander Bogomolny