Here is a multiple choice math question that might appear on a test of intelligence or scholastic aptitude.

X and y are two integers such that x² + y = 704369 (in other words X squared + Y = 704369). Which of the following values could be valid values for y?

1. 216
2. 447
3. 571
4. 882
5. All of the above
6. None of the above

It isn't difficult to see how this question could be answered. Subtract the suggested value for y from 704369. Then take the square root of the result and see if it is an integer. But unless you have a calculator that can take square roots, this becomes a tiresome exercise instead of a puzzle. We are looking for an easy way to check each possible answer without having to take a square root or having to guess a value for x and square it.

The real puzzle here is to come up with an insight that will allow you to answer the question without any difficult calculations and without a calculator. In other words, do it quickly, in your head.