Problems

There are \(57\) buttons placed evenly around a circle, each with a light underneath it. Pressing a button changes the state of its own light and also the lights under its two neighbouring buttons. Changing state means that a light which is on becomes off, and a light which is off becomes on. If all the lights start off, what is the smallest number of presses that need to be done so that the entire circle lights up?

Without using any wolves, show that Robinson’s goat can only graze shapes that are convex (that means, whenever you pick two points inside the shape, the whole line between them also lies inside). But if Robinson is allowed to use as many wolves as he likes, this restriction disappears. Show that in this case, he can make the goat graze in the shape of any polygon at all.

In a field, there are two pegs, \(A\) and \(B\), placed \(15\) metres apart. Each peg has a small ring on top, and a rope can slide freely through these rings. You have one rope and two goats that want to graze the grass, but they will fight each other if they can both reach the same spot.

The rope may be arranged in any way you choose: it could pass through both rings, only one, or neither, depending on your setup. For what lengths of rope can you arrange things so that the goats cannot fight each other?

Fred starts running from point \(A\) and must reach point \(B\). On the way, he has to touch the fence shown as a straight line in the figure. It doesn’t matter where he touches the fence, as long as he does. What is the shortest path he can take?

Robbie and Paloma are playing football on a large, flat field. They both run at the same speed toward the ball. Where could the ball have been if Paloma reached it first?

Oliver starts running from point \(A\) in the figure below. He must touch line \(H\), and then line \(L\). What is the shortest path he can take?

Two ants move along the same straight line and can only move back and forth along it. The first ant moves twice as fast as the second. Where could you place a breadcrumb so that the slower ant reaches it first?

Two ants, Muffy and Chip, start on opposite sides of a circle. Muffy moves twice as fast as Chip, and both can only move along the circle’s edge. Where could a breadcrumb be placed so that they both reach it at the same time?

Two points, \(A\) and \(B\), lie between two lines \(L\) and \(H\). Draw the shortest path from \(A\) to \(B\) that touches \(L\), then \(H\), and finally reaches \(B\).

Ignat is playing pool and places a ball at point \(A\), as shown below. Can he bounce ball \(A\) off a ball placed at position \(B\), if the ball can only bounce off the top and right edges of the pool table?

When a ball bounces off the edge, it does so just like light reflecting in a mirror: the angle at which it hits the edge is the same as the angle at which it bounces away.