Henry Ernest Dudeney/Modern Puzzles/115 - The Carpenter's Puzzle/Solution

by : $115$

 * The Carpenter's Puzzle

Solution
There is only one general situation in which it is possible to dissect a rectangle into two pieces using this specific technique in order to make a square.

That is when the sides of the rectangle are in the proportion of consecutive square numbers.

Thus for a rectangle whose sides are in the ratio $n^2 : \paren {n + 1}^2$, a cut of $n$ steps is used to dissect it into a square whose sides are $n \paren {n + 1}$.