Why calculating XOR of consecutive values can be simplified?

Question

I was trying to calculate integer xor of 0..n. I named the function xored(n).

Note that in examples below ^ does not mean power but integer xor (like in C or Java language)

So, xored(0) = 0, xored(1) = 0 ^ 1, xored(2) = 0 ^ 1 ^ 2, xored(3) = 0 ^ 1 ^ 2 ^ 3, etc...
Or, if you prefer recursive definition:
xored(0) = 0
xored(n) = n ^ xored(n-1)

First, I created a naive implementation that that loops through the values and xors them together.

Then, I noticed a funny thing. I noticed a pattern with my results.

All you need to do is to calculate n modulo 4 (call it m) then:
if m = 0, xored(n) = n
if m = 1, xored(n) = 1
if m = 2, xored(n) = n + 1
if m = 3, xored(n) = 0

My question is. Why does it work? How can this be proven mathematically? I am not good at proving theorems...

Hint: 0, 1, 2, 3 in binary are 00, 01, 10, 11...... – Brian Cheung Aug 25 '15 at 19:18 — Brian Cheung, Aug 25 '15 at 19:18
That I know. But how does this help me? – Paweł Aug 25 '15 at 19:19 — Paweł, Aug 25 '15 at 19:19
0^m00=m00, m00^m01=1, 1^m10=m11, m11^m11=0...... – Brian Cheung Aug 25 '15 at 19:21 — Brian Cheung, Aug 25 '15 at 19:21

score 0 · Answer 1 · edited Apr 13 '17 at 12:21

I think some of the proofs are based on observations and generalizing the patterns. It helps at many places where you can't prove things using a theorem but things can be sorted out using observations.

Euler's Polyhedral Formula $ V(vertices) + F(faces) - E(edges) = 2$ is such an example. Euler did not give a proper proof (though it was proved in many ways afterwards), this result inspired huge advances.

EDIT: There are a lot of threads with the same question and everyone solved it using mere observation. One of the threads can be found here.

Why calculating XOR of consecutive values can be simplified?

1 Answers1