I played **ultimate** tonight, and we usually keep score with shoes. Our normal scoring system is to count in base 5. Tonight, I tried to use binary, but at half-time I switched back to base 5 when most of our team struggled to read our score quickly.

I took some pictures of the arrangement of shoes during the game (when I wasn’t playing).

I can imagine some investigations could be made out of these photos.

- Given the numbers associated with each photo, try and determine how to count in this number system,
- More challenging: From these photos, try and determine the missing numbers.

If you want a project that might take a while:

- Design a scoring system using shoes. It should be easy to maintain and not require too many shoes.

I can understand the base 2 system. With the base 5 system is it the case that you can place each shoe in 5 different positions that map to the values 0 through 4? If so, what’s the mapping?

In the "base 5" system for scoring in Ultimate, a show facing sideways has value 5, and a shoe perpendicular to that has value one, and you literally just add the values together. It’s not really base 5 (since 25 would be represented with five horizontally placed shoes) but it’s related.

All of the pictures above are in base 2.