U(7)

A game of dice, speed, and cyclic groups.

Play

Learn

How to Play

Take turns rolling the die, and keep track of the cumulative product (mod 7).

If the product becomes a 1 with the current roll, be the first to grab the paper, and you get a point.

But BE CAREFUL grabbing the paper - if the product is NOT 1, you LOSE a point!

First player to reach 2 points wins.

Background

Define U(n) or (Z/nZ)^× as the set of positive integers less than n that are relatively prime to n. This forms an abelian group under multiplication mod n (so if we keep multiplying values, we will always get a product in this set).

If n is prime, U(n) is cyclic and consists of all positive integers less than n. For instance, U(7) =
{1, 2, 3, 4, 5, 6} (all the numbers on a game die).

For this game, note that every element of U(n) has an inverse (i.e., for every number a in U(n), there is a b in U(n) with ab = 1).