Like the number 15, all odd numbers can be written as the sum of two consecutive integers.
For any odd number k, we know that k-1 and k+1 are even, also ( k-1)/2 and ( k+1)/2 are consecutive integers, so we can write k=( k-1)/2 + ( k+1)/2.

All numbers of the form N = pk where p and k are integers, and k is odd and greater than one, can be written as the sum of consecutive integers.

As before we use the integers ( k-1)/2 and ( k+1)/2 but this time we have to add 2 p consecutive integers, p of them less than ( k/2) and p of them greater than ( k/2), so that their mean is k/2 and their sum is (2 p) x ( k/2) = pk. For example N = 44 = 4 x 11 can be written as the sum of 8 consecutive integers, 4 less than 5.5 and 4 greater than 5.5 so that the sum is 8 x 5.5 = 44.

44 = (11 - 7)/2 + (11 - 5)/2 + (11 - 3)/2 + (11 - 1)/2 + (11 + 1)/2 + (11 + 3)/2 + (11 + 5)/2 + (11 + 7)/2 = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.

This construction works for all numbers of the form N = pk where p and k are integers and k is odd and greater than one, but the consecutive sum may include some negative integers. If this happens then a string of terms reduces to zero giving fewer terms in the final expression. By this method, 12 = 4 x 3 = (-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 = 3 + 4 + 5 giving three terms in the final expression rather than eight.

If a number is a power of 2 then it is impossible to write it as a sum of consecutive integers. Consider the sum of consecutive integers starting at ( q + 1) and ending at p. There are ( p - q) terms in this sum. The mean of the terms is ( p + q + 1)/2 so we can write:

( q + 1) + ( q + 2) + ...... +( p - 1) + p = ( p - q)( p + q + 1)/2.

As ( p - q) and ( p + q + 1) cannot both be even, this means that the sum of consecutive integers must represent a number which has at least one odd factor (other than the factor 1) so a power of 2 cannot be written as a sum of consecutive integers.

We use the fact here that if p - q is odd then p + q is odd and p + q + 1 is even, or if p - q is even then p + q is even and p + q + 1 is odd.

This solution may be particularly helpful to Melanie Kemp of West Flegg Middle School, who sent in a very well reasoned, part solution to the problem.

, Paul Latham and Matthew Tuddenham sent in a correct solution and again the above reasoning may help you to understand why it works.