Back in high school I discovered this by accident (yes, I was really bored!). I suppose it's nothing new, but it turns out that this works for more than simple squares and cubes:

Given any sequence of numbers, keep finding differences of differences until you hit a constant; the number of iterations needed is the maximum exponent in the formula that produced the numbers. That is, this works even if there are other terms, regardless of whether any or all terms have coefficients other than 1.