I have NEVER seen this problem, although it wasn't that hard to figure out. Whoever discovered this problem, I have to hand it them, they are pretty creative.
Here's the how and why of this problem:
First, what happens when you square a triangular number? In general a triangular number is n(n+1)/2, so if it is squared it is [n(n+1)/2]^2 which is the formula for finding out the sum of sequences in this form: 1^3 + 2^3 + 3^3 + ... + n^3. So if we are given a problem of this sort:
a^3 = x^2 - y^2, where x and y are triangular numbers then that means we want to know:
What does this mean? Basically, whatever cube is to the left of the equation, we know that x = that same number and y=that number minus 1. But remember these are triangular numbers. So find the triangular number that corresponds to that number.
I know this sounds REALLY confusing, so let's look at some examples to help us out.
Ex [1] 6^3 = x^2 - y^2, where x,y > 0 and x,y are triangular numbers, find x.
We will focuse on 6, since that is the number being cubed.
x = 6th triangular number and y = 5th triangular number.
x = 6*(6+1)/2 = 3*7 = 21.
The answer is 21.
*Notice that y=5*(5+1)/2 = 15. So to check we can see if 6^3 = 21^2 - 15^2. This is 216 = 441 - 225 which is correct.
Ex [2] 64 = x^2 - y^2, where x,y < 0 and x,y are triangular numbers, find y.
We know that 64 is 4^3, so we will focus on 4.
x = 4th triangular number and y = 3rd triangular number.
y = 3*(3+1)/2 = 3*2 = 6.
Since y<0 the answer is -6.
*Notice that x = 4*(4+1)/2 = 10. Since x<0, x=-10. To check we can see if 64 = (-10)^2 - (-6)^2. This is 64 = 100 - 36 which is correct.
