Square Cross Puzzle

How did Ramanujan solve the STRAND puzzle?
Mathologer

I started watching this video the yesterday. I got as far as the square cross puzzle (timestamp 1:45 to 2:20), and that intrigued me. I stewed on it overnight and this morning I came up with this analysis.

Square Cross Problem

We start with a drawing of a cross. The cross is composed of five squares. The center of the cross is a square. Attached to each side of the central square is another square of the same size.

Our mission, should we choose to accept it, is to cut the cross into five pieces and then reassemble those pieces into two crosses with the same shape. These two new crosses are the same size as each other.

Since we are making two crosses from the original larger cross, each of the smaller crosses must be one half of the area of the original larger cross.

If the length of the side of one of the component squares of the larger cross is one, a little algebra will show us the the length of the side of one of the component squares of the smaller crosses must by the (2^0.5)/2 (the square root of two divided by two).

Cutting each of the component squares of the original square on both diagonals will give us 20 right isosceles triangles. Two of those triangles joined together along their hypotenuse will give us a square the same size as one of the component squares of the smaller crosses. Two triangles for each of five squares requires ten triangles to make one smaller cross. Two times ten is twenty, the number of triangles we got by cutting up the larger cross.

This is the brute force method. It shows we can cut up the larger cross into smaller pieces and reassemble them into two, smaller, crosses. However, we have twenty pieces, not five, so we do not have a solution to the stated problem.

The trick is to find cuts that do not need to be made so that when we cut up the larger square we only have five pieces. Now I'm thinking a computer program could make short work of this if I can just figure out how to encode it.