1. ## flying ducks

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?

2. ## Re: flying ducks

You are looking for numbers of the form $\binom{a}{2}+\binom{b}{2} = \binom{c}{2} < 100$ where $a,b,c$ are integers greater than one.

This can be written: $a(a-1)+b(b-1) = c(c-1) < 200$

This is known as a Diophantine Equation. There are methods to solving these, but the math is extremely complex, and in this instance would likely take more work than a simple guess and check method. In mathematics, you typically look for the method that requires the least computation. So,

Here is a list of all triangle numbers up to 100:

1,3,6,10,15,21,28,36,45,55,66,78,91

Checking sums, you find many answers that work (I don't see how 36+55=91 is any better than any of these):
3+3 = 6
6+15 = 21
10+45 = 55
15+21 = 36
21+45 = 66
36+55 = 91

If the question asked for the total to be as close to 100 as possible, then 36+55=91 would be the solution, but as you posed it, there seem to be six valid answers.

3. ## Re: flying ducks

Thanks a lot for the help.