Also because we end up back at zero when ever n % 4 == 3, there is some flexibility about the starting point. I wrote 1 to n because that is what the article used, but it would be mathematically cleaner to start at zero which actually changes nothing because XORing with zero does nothing. And we do not have to start at zero at all, we can start at any number divisible by four and less than n because the running XOR sum will become zero just before each multiple of four. So XORing together 0...n, 1...n, 4...n, 8...n or generally 4k...n will give the same result. The explanation part looked at one cycle starting at 4k and ending at 4k + 3 with the running XOR sum being back at zero. Maybe this would have been the less confusing explanation, just using 4k instead of using n again with the constraint that it is divisible by four.