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▲ betasilly a day ago

Another interesting fact is that each time you make the xor of four consecutive numbers, beginning with an even number, the result is zero. Example in J.

  xor =: (16 + 2b0110) b.
  f =: 3 : 'xor/ y + i. 4'
  f"0 ] 2 * 1 + i. 100
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Summing a hundred millions: +/ f"0 ] 2 * i 100000000 gives zero (it takes a few seconds). So it seems the stated property holds for every even n.

▲danbruc a day ago | parent [-]

Yes, because XORing two consecutive integers only differing in the least significant bit yields one.

  xxxxxxx0 ^ xxxxxxx1 = 00000001
Doing this twice with four consecutive numbers then also cancels the remaining one. That also means that you do not have to use consecutive numbers, you can use two arbitrary pairs

  2m ^ 2m + 1 ^ 2n ^ 2n + 1
and for example

  16 ^ 17 ^ 42 ^ 43
should be zero.