I believe this problem has long since been solved, and it came up several "sit-down meetings" ago, but Z's comment to a baseball post reminded me that we do have some cool math geek readers here.
Consider infinitely many rows of tiles in a triangle pattern: The first row has one tile, second has two tiles, nth has n tiles.
Which positive integers can('t) be formed as the sum of two or more consecutive rows of tiles?
(Henceforth "number" describes only positive integers.)
Two consecutive rows can form any odd number greater than 1. Any odd number is expressible as 2i+1 for some i, and if i > 0 then you use the ith row and the (i+1)th row.
Three consecutive rows can form any proper multiple of 3 because i + (i+1) + (i+2) = 3i + 3 = 3(i+1).
More generally any odd number of consecutive rows can form any multiple of that odd number greater than half the square of that odd number.
Four consecutive rows can form any 2mod4 number >= 10. Six consecutive rows can form any 3mod6 number >= 21, eight consecutive rows can form any 4mod8 number >=36.
More generally, any number that can be expressed as the product of an even number and an odd number can be represented by consecutive rows. If 2E > O then use O rows whose average is E. If 2E < O then use 2E rows whose average is O/2. (For a quick 10 points why will you never see 2E == O?)
We've covered every odd number greater than 1 and every even number expressible as the product of an odd and an even. The only even numbers not expressible as the product of an odd and an even are powers of 2.
An exercise for the reader is to prove that NO power of 2 can be expressed as the sum of N consecutive integers.
Posted by Matt Bruce at August 10, 2007 01:53 PMIt's actually pretty easy to prove, based on the fact that no power of 2 has an odd factor.
For any odd N, the sum of N consecutive integers is a multiple of N (it's N times the middle integer).
For any even N, the sum of N consecutive integers is a multiple of the sum of the middle two integers. Since exactly one of those two integers is odd, that sum is odd.
Posted by: me at August 10, 2007 09:27 PM