Two regular hexagons share two vertices. What fraction of the area is purple?

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## Solution

The purple fraction is 13/29.

First, relate the diagonal connecting the shared vertices to the respective side lengths a and b. This gives b/a=√3/2 and therefore the smaller hexagon has area B that is 3/4 the area A of the larger one: B=3A/4.

Now splitting the purple area in two by this diagonal, it is easy two see that the two parts are A/6 and B/2. Thus, this overlap region is 13A/24 or equally 13B/18. We find that X=11A/24 and Y=5B/18=5A/24.

The required fraction is 13/(13+11+5)=13/29.

## Poem

We can go in winter

Or also in summer

In boxcars travelling in coal dust and cinders

They sometimes are loaded with clothes or computers

Some men sold a boxcar filled with wheat flour

They made a good deal in less than 1 hour

They were so young and fun

They waved to the sun.