Four squares inside another square. What fraction is yellow?

Scroll down for a solution to this problem.

## Solution

The yellow square covers 1/5 of the total area.

First, scale the diagram such that the red squares have unit sides and call the yellow side x. Comparing bottom and left side of the large square, it is straightforward to fill in the lengths as shown, also using congruency of the two left-side triangles.

Now consider the small triangle in the middle. It is similar to the larger ones and this gives (x-1)/1=1/x. This has one positive solution x=(1+√5)/2, also know as the Golden Ratio. The required fraction is x^{2}/(2+x)^{2}=1/5.

## Poem

The corner stone

Not a heart being alone

The stony heart is speaking

Cannot stop skipping

Cause the lonely girl is crying

She realizes she is suffering

Stones are hard and bleeding

And secrets conversing

Some of them are holding

And try to hide something

Stones are fighting