I didn't know this lovely puzzle until I stumbled across today it on the estimable Numberphile YouTube channel.

A mouse is swimming in a round pond. A cat circles the pond, but won't enter the water. This is my drawing, I think I have a better cat than Numerfile TBH.

If the mouse reaches the shore and is not intercepted by the cat at that point, it can run quickly and escape the cat. The cat can run 4 times faster than the mouse can swim.

Can the mouse always escape if it follows the right strategy?

The answer is presented by Numberphile in a really illuminating way. Just Google Numerphile Game of Cat and Mouse.