Given positive, integer N, what does this algorithm produce?
let S and D = N
let D = ⌊D/2⌋
subtract D from S
produce S as the result
⌋ represents the integer part of x
For non-programmers: start with a positive, integer number (say, 19). Divide it by 2, discarding remainders, until you get to 0. (In this case, you'd get 9, 4, 2, and 1.) Sum all the quotients. (9+4+2+1=16.) Subtract the sum from the original number. (19-16=3.) What's the result?
First, consider the numbers in binary system. N might be, say,
11010 or 26 in decimal. When D is halved and rounded down, it is
like removing the last digit of the binary form. So first it
would become 1101, then 110, 11, 1, and 0.
This means that if there is a 1 in, say, the 4th digit from the right,
a total of 111 will be subtracted from S before the one
disappears. Naturally, 1000-111 is 1, as is 1000000-111111 and
10-1, etc. Therefore, the resulting S will be the sum of the
digits of N, when N is in binary form.
Posted by Tristan
on 2005-01-14 23:00:39