Solution: The number x has 6 digits, so it must be of the form

where A, B, and C are the digits of the number. If we wished, we could expand x as

where A, B, and C are integers less than 10. We require only that

If it were possible for A to be 0, then we would have a difficult (impossible) task ahead of us. However, A cannot be zero because then x would be less than 100,000. With our restriction on the relative sizes of A, B, and C, this means that B cannot be zero either.

Now we need three positive integers less than 10 which satisfy Pythagoras' theorem. The numbers 3, 4, and 5 satisfy the above conditions, so

is a valid solution. Notice that we've not yet claimed that this is the only valid solution (even though we suspect it is), but have simply taken a guess at one valid solution. To demonstrate that this is the only solution would require either a long trial-and-error procedure or a very sophisticated proof.