On the art of cutting squares

Geometry Level 2

Given $n$ squares of side lengths $a_1, a_2, \cdots, a_n$ , is it possible to cut them in a way that we can rearrange all the pieces to obtain a single square? (Taken from here )

When

n

is even Always When

n

is odd Never

1 solution

Rodrigo Girão Serrão
Aug 20, 2018

Yes you can.

A solution can be read here .

By induction, we only need to care about the case $n = 2$ . If it is true that, given any two squares, we can cut them into one single square, then we can do it for any $n$ : just lay all the squares side by side (call them $A_1, A_2, \cdots, A_n$ ) and start with the two to the left, creating a new square $B_1$ . Then take $B_1$ and $A_3$ to create $B_2$ . Keep going until you create $B_{n-1}$ from $B_{n-2}$ and $A_n$ .

To show that it is possible for two square we refer to this explanation which has figures (and hence is easier to understand).

On the art of cutting squares

1 solution

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