Nimbers

To add two Nimbers you follow slightly different rules from normal addition. You still have the expected properties that:

• x + y = y + x
• x + (y + z) = (x + y) + z

but we have some additional rules including:

• x + x = 0 for any x.

To add numbers that are not equal, we break the number up into a sum of powers of two always taking out the biggest possible power of two first. E.g., 10 = 8 + 2, 7 = 4 + 2 + 1. Because we start with the largest possible power of two, there can never be two of the same number in this decomposition or else we were not aggressive enough. We then use the fact that x + x = 0 to cancel pairs and finally the sum we get is given by adding all the powers of two that did not have pair.

For example if we want to add 10 and 7 we write 10 = 8 + 2, 7 = 4 + 2 + 1. The pair of 2's cancels and so we are left with 15 = 8 + 4 + 2 + 1.

Similarly to add 6 and 3 we get 7 = 4 + 2 + 1, 3 = 2 + 1. The 2's and 1's cancel leaving 4.

Lastly adding 1, 2 and 3 we get 1 = 1, 2 = 2, 3 = 1 + 2. Everything cancels to leave 0.