**From New Scientist #2326, 19th January 2002** [link]

I have a rectangular piece of paper which I have folded twice and then unfolded again. The first fold was as shown:

In that figure the area of each of the three triangles is a perfect square number of cm², and each area is less than 100 [cm²].

The second fold was through X and parallel to the first. When unfolded the two creases divided the rectangle into three regions, namely two triangles and a pentagon. The area of each was again a perfect square number of cm².

What is the area of the rectangle?

[enigma1170]

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Enigma 1402.I found the analysis a bit fiddly on this one. The maths isn’t that hard, but it is easy to make a mistake. I used

SymPyto do some of the expression manipulation for me.Suppose we are given the square numbers for triangles

AandB, 0 <a<b< 10.Now, if triangle

A(areaa²) has widthxand heighty, then:Triangle

B(areab²) is (mathematically) similar to triangleA, and the sides are in ratior=b/a. Its hypotenuse is the bottom edge of the paper, which has lengthx+ry. So:Substituting for

randxand solving fory:and it follows for triangle

C(areac²) that:For the lower triangle

C, where the angle in the right hand corner isφwe have:and, in the second diagram, the triangle

Dhas area:and the region

Ehas area:So, given

aandb, we can work outc,d,eand check that they are all integers from 1 to 9.This Python program looks at all the possibilities in 32ms.

Solution:The rectangle has area 90 cm².Here are some diagrams, which should help explain things:

The computed values are:

My initial attempt at the maths got too complicated until I realised that the tangent formula for double angles provided a simple solution. A nice challenge.