## Alternating Squares

#### Problem

A grey square has a white circle inscribed and a grey square inscribed in the circle.

What fraction of the diagram is white?

#### Solution

Area of outside square = 2`r` 2`r` = 4`r`^{2}

Area of inside square = ½ 4`r`^{2} = 2`r`^{2}

Area of circle = π`r`^{2}

Area white = π`r`^{2} 2`r`^{2} = `r`^{2}(π 2)

Hence, fraction of the diagram white = `r`^{2}(π 2)/4`r`^{2} = (π 2)/4 0.285

If another white circle was inscribed in the inside square, what fraction would be white now?

What would happen if white circles and grey squares were recursively inscribed? Would the white area tend towards a limit?

Problem ID: 119 (May 2003) Difficulty: 2 Star