## Teaser 2864: Sequence of squares

**From The Sunday Times, 13th August 2017** [link]

I started with a rectangle of paper. With one straight cut I divided it into a square and a rectangle. I put the square to one side and started again with the remaining rectangle. With one straight cut I divided it into a square and a rectangle. I put the square (which was smaller than the previous one) to one side and started again with the remaining rectangle. I kept repeating this process (discarding a smaller square each time) until eventually the remaining rectangle was itself a square and it had sides of length one centimetre. So overall I had divided the original piece of paper into squares. The average area of the squares was a two-figure number of square centimetres.

What were the dimensions of the original rectangle?

[teaser2864]

## Jim Randell 1:00 pm

on21 May 2020 Permalink |If we start with the 1×1 cm square and replace the removed squares, we find the sequence of sizes of squares is:

i.e. a Fibonacci sequence. [ @Wikipedia ]

So we can start with the two 1×1 cm squares and build up the original rectangle square by square, until find one where the mean area of the squares is a two digit integer as required.

This Python program runs in 82ms.

Run:[ @repl.it ]Solution:The original rectangle was 13 cm × 21 cm.So in total 6 cuts were made, producing 7 squares.

The total area of the 7 squares is 273 sq cm, so the mean area is 39 sq cm.

Manually:

If:

Then we can calculate:

And the answer is apparent.

If we draw a quarter circle through each square we can make a

Fibonacci Spiral:LikeLike