Note: The previous puzzles have been reruns of puzzles that I created for the Puzzle of the Week Contest that I run at Thompson Rivers University in Kamloops, British Columbia, Canada. Now that the academic year has started, counting this puzzle, the next ten puzzles will be new for that contest.

Triangular numbers are numbers that are the sum of the integers from 1 to some positive integer. Call the function T(*n*). T(1) = 1, T(2) = 1 + 2 = 3, T(3) = 1 + 2 + 3 = 6, and so on.

Some numbers can be expressed as T(*a*) – T(*b*) for some *a* and *b*. For example, 12 because T(5) – T(2) = 15 – 3 = 12.

Prove that for all integers *n* ≥ 2, that there are values *a* and *b* such that T(*a*) – T(*b*) = *n*.

Hint: There is a very simple solution which you may find if you look at the problem right. Drawing diagrams may help.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. **The deadline is Wednesday, September 25, 2013 at noon Pacific Time.** I will post the answer shortly after.