Hello,

I came across a problem that no one can seem to answer without some guess work, which I severely dislike.

A group of ducks are flying across the sky. Ducks always fly in equilateral triangles with 1 duck at the front, 2 behind him, 3 behind the other two and so on and so forth. A hunter shoots at the ducks and misses, but the sound of the gun is enough to disrupt their flight, so they break off into two equilateral triangles (not necessarily congruent). If the number of ducks that are flying are less than 100, what how many ducks are flying?

The answer is 36+55=91, which are all triangular numbers.

Is there any formula where I can add two different triangular numbers to get another triangular number?