This week the question is: Can You Bounce a Ball Like an AI?

… suppose you have a unit square that’s rotating about its center at a constant (nonzero) angular speed, and there’s a moving ball inside. The ball has a constant (nonzero) linear speed, and there’s no friction or gravity. When the ball hits an edge of the square, it simply reflects as though the square is momentarily stationary during the briefest of moments they’re in contact. Also, the ball is not allowed to hit a corner of the square—it would get jammed in that corner, a situation we prefer to avoid.

Suppose the ball travels on a periodic (i.e., repeating) path, and that it only ever makes contact with a single point on the unit square. What is the shortest distance the ball could travel in one complete loop of this path?

And for extra credit:

Again, you have a rotating unit square and bouncing ball inside.

By now, you’ve hopefully found the shortest repeating path for which the ball makes contact with a single point on the square. Let’s call this path length L₁.

The next shortest repeating path for which the ball makes contact with a single point on the square has length L₂. To be clear, L₂ > L₁.

What is the length L₂?

Highlight to reveal (possibly incorrect) solution:

Spreadsheet.

A few tests lead me to believe, the only valid solutions are loops coinciding with regular polygons within a unit circle. The point on the square will be in the middle of one of the sides.

The first polygon is simply the loop of going 2 in one direction and 2 back. L₁ = 4.

The next polygon is a triangle. Each side of this triangle is also a long side of a triangle, where the centre of the unit circle is the 3rd vertex. 2 sides have the length 1. The angle at the centre is 360°/3 = 120°. The angles at the other vertices are (180°-120°)/2 = 30°. The length of the unknown side, x, satifies x/sin(120°) = 1/sin(30°) <=> x = √3. The perimeter of the triangle is 3√3, about 5.196.

The spreadsheet continues this trend. The perimeter of the polygons grow towards 2π.

Oh wait. While a unit circle has radius 1, diameter 2, a unit square simply has width 1, right? Sigh. — All results above should be divided by 2.