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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