Three of these four numbers can be written as one square number added to another square number (using 1, 4, 9, 16, 25, … ). Which one cannot? 13, 25, 34, 43

Test each in turn. 13 = 4 + 9, 25 = 9 + 16 and 34 = 25 + 9, so those three work. For 43, take away each square in turn: 43 − 1 = 42, 43 − 4 = 39, 43 − 9 = 34, 43 − 16 = 27, 43 − 25 = 18 and 43 − 36 = 7 — not one of those leftovers is a square, so 43 cannot be written that way. The traps: 25 stands out because it is itself a perfect square, but 9 + 16 settles it; 34 is the only even number in the list, another surface difference that has nothing to do with the rule; 13 is the smallest of the four and the one people test least carefully.