Can You please tell proof of second method

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The cumulative xor of numbers 1...N shows this wonderful property where it repeats according to the pattern mentioned in the article. I guess the only explanation I can think of is that the bits get cancelled when (N+1)|4 and the rest just follows as:

0 \oplus x=x, \forall x \in \mathbb{B}

i.e.,

0 | 0

\oplus \oplus

0 | 1

\rule{0.6cm}{0.4pt}

0 | 1

Here’s another link with some more explanation regarding the topic. I’m not sure whether you’ll find a rigorous proof of this though. Hope this helps you.

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