If we have a sequence from 0 to N in binary format then the number of 0 and number of 1s in the least significant bit have a balance i.e. if N was even then there is one 0 more than how many 1s are there. If it is odd the number of 0s and 1s are the same.
0000 0001 0010 0011
There are 2 0s and 2 1s in the first binary column (N = 3 i.e. odd)
0000 0001 0010 0011 0100
There are 3 0s and 2 1s in the first binary column (N = 4 i.e. even).
This makes sense since each number is basically the previous +1, so each addition either adds a 1 or a 0 to the last column and since this is about all the numbers from 0 to N this equation is there.
What is not straightforward to me to understand intuitively is that if I split these numbers to 2 sets, i.e. odds (or those with the last bit set to 1) and evens (those with the last bit set to 0) the same equation holds.
The same also holds for all further subdivisions of those with the second column 0 vs second column 1 etc.
Could some give some insight why this further and further subdivision keeps having this condition hold?