### 538 Riddler: Puzzle of the Picky Eater

#### 22 June ’16

I found it easiest to think about this problem in terms of polar coordinates. The furthest point that we can eat along trajectory is the following:

Plotting this, it forms this weird, rounded rectangle shape:

I integrated the above equation to get the area. Unlike a regular integral where each area is a rectangle (Reimann sum), each incremental integration area is a circular sector with . Putting this all together:

Extending this logic, I also derived an expression for the area eaten for a regular polygon with n sides:

As n goes to infinity, the area that the picky eater eats becomes a circle with a radius half that of the sandwich, making the area that is eaten equal to 25%. This is the most efficient shape for the picky eater.