A square contains a smaller square. Their vertices are cyclically connected as shown, forming four triangles. What is the minimal green area fraction?

Scroll down for a solution to this problem.

## Solution

The minimal green fraction is 1/2.

First it is seen that the problem satisfies translational symmetry of the small square centre, since the sum of the triangle heights is constant. We translate it such that the two squares are concentric and the figure has fourfold rotational symmetry.

We scale the large square to 1. Naming the vertex in the first quadrant (x,y), we can calculate the small square area as half the square of the diagonal, which evaluates to 2(x^{2}+y^{2}).

In the concentric case, all four triangles are congruent and have area (1/2-x)/2. Therefore the green area sums to A(x,y)=2(x^{2}+y^{2}+1/2-x). To find the extremes, we set dA/dx and dA/dy to 0. This gives x=1/2 and y=0. This is indeed a minimum and it evaluates to 1/2.

Find here a beautiful animation by Ignacio Larrosa Cañestro and notice how counterintuitive it looks as it approaches the minimum.

## Poem

The snowdrop every year, still

Is growing and glooming with will

First greeting from the spring

It’s light pure and white shining

After a long sleep under snow

The little flower speaks to our hearts, it knows

Full of innocence

Opening humbly,

The snowdrop is an evidence