Cutting Squares

Here is a problem:

Given N > 1 squares of arbitrary sizes. Is it always possible to dissect the squares into pieces that will combine (without overlapping or holes) into a bigger square?

For N = 1, the question is vacuous. For N = 2, we actually have several solutions. 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

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

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