63 is a polite number because 31 and 32 which are consecutive add up to 63. However 64 must be impolite since consecutive integers cannot be formed with 2 or 3 consecutive numbers. While 64/4=16, 14, 15, 16 and 17 do not add to 64. I though this may apply to all powers of two- so I tested the first three of them below:
2- the only numbers here could be 0,1. Two ones would form two, but these are the same number so not consecutive
4- 1,2,3 are the available numbers. 1+2+3=6. 1+2=3. 2+3=5. These are the only available consecutive numbers for this set.
8- 1,2,3,4,5,6,7, are the available numbers here. 1+2=3. 2+3=5. 3+4=5, 4+5=9. ( It is evident here that the further numbers will be in excess of 8)
We can see that after 2 ( which must be impolite since it only has one number smaller than it) the powers of two are halfway through (in for number eight 3.5 and 4.5 would add to 8 but these are not integers). I have algebraically proven this in my image.
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I hope this link connects to my image.