## PERFECT SQUARES

Three of the following four numbered statements are TRUE. One of them is FALSE. What is the value of x?

1. x is a perfect square.
2. x+37 is a perfect square.
3. x+98 is a perfect square.
4. x is odd.

1. x is not a perfect square so this statement is FALSE.
2. x+37 = 900 = 30 squared -- TRUE.
3. x+98 = 961 = 31 squared -- TRUE.
4. x is odd -- TRUE.

To show that there is no other answer we must try to (a) show that statement 1 is false with another answer that keeps statements 2, 3 & 4 true or (b) find solutions when statements 2, 3 or 4 are false with the others true.

We start off by making the assumption that statement 1 is true; i.e. that x is a perfect square, and try to prove that it is or is not.

To find solutions to equations like b*b + n = t*t we note that
n = t*t - b*b = (t+b)*(t-b) which requires finding all the factors of n. For statements 1 & 2, n = 37 which only has factors of 1 and 37.
For statements 1 & 3, n = 98.
For statements 2 and 3, n = 98 - 37 = 61.

For statements 1 & 2, the mean value of (t+b) and (t-b) is t which is 19.
This makes b = 18 and x = 18*18 = 324.
This makes statement 4 false so this would be a solution if statement 3 were true.

If x = 324, statement 3 is false because 324+98 = 422 which is not a perfect square. So x = 324 is not a solution to the main problem.

If we investigate the possibility that statements 1 and 3 are true we come up against the fact that the number 98 has factors that sum to an odd number and therefore the value of t cannot be an integer. So there is no value of x that satisfies statements 2 and 3.

That leaves the only possible answer that statements 2, 3 and 4 are true and 1 is false. In this case we must use an offset of 37 to make our algebraic symbols match the problem so that n = 98-37 = 61. The only factors of 61 are 61 and 1 that leads to t = 31 (the mean value of 61 and 1) and b = 30 with our offset "x" = 900.
But we must now correct for the offset of 37 so x = 900-37 = 863.
Checking the result we see that x + 37 = 900 which is 30*30 and x + 98 = 961 = 31*31.
This makes statements 2 and 3 correct. 863 is not a perfect square so statement 1 is false. x is odd so statement 4 is true. Thus the value of x is 863.

I received five correct answers. They were from John Bownas, John Stafford, Michael Hodge, Richard Burkill and Clem Robertson (who won the draw). Well done all.

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