Problem #WSP-000009

Problem

Let us define XOR (or addition mod 2). XOR is defined for 0 and 1 only. Here is a table recording the values of XOR:

Now we define the important concept of nim-sum. Given two natural numbers $x$ and $y$, we first convert them into binary representations and then compute XOR on individual digits. The resulting number, denoted $x\oplus y$, is the nim-sum of $x$ and $y$. Here is an example.

This is simply saying $22\oplus 5=19$. Note that $22=(10110{)}_2$ and $5=(00101{)}_2$.

Verify $(x\oplus y)\oplus z=x\oplus (y\oplus z)$, so we can speak of $x\oplus y\oplus z$ with no ambiguity.