Hopscotch was originally a training exercise for Roman infantry. These frogs don't know that history. They are just interested in jumping about on their lilly pads. Each frog jumps a different distance:

This means that when the green frog jumps, it only jumps to a nearby lilly pad and when the yellow frog jumps, it jumps over three lilly pads and lands on the fourth.

Arrange these four frogs on the five lilly pads so that all of them can jump at the same time every minute, and never share a lilly pad? The above arrangement doesn't work because the blue and the yellow frogs will be jumping to the same lilly pad:

The following arrangement also doesn't work because the red frog won't have a lilly pad to itself after jumping:

Extensions:

Arrange 5 frogs with jumps 1,2,3,4 & 5 on six lilly pads.

Is it always possible to arrange N frogs on N+1 lilly pads?

Can you find the number of Lilly Pads required for N frogs which jump 0, 1, 2... N-1? *

What happens if each jumping frog owns two lilly pads and refuses to let other frogs touch them? Is it possible that N frogs can be happy on 2N lilly pads? *

For example:

is a solution for the four frogs which jump 1, 2, 3, and 4.

*I am working on this problem. Not all numbers are possible.

Download PDF