The original puzzleA young girl, bearing a basket of apples, must, at each of three gates, give to the keeper of the gate half of her apples plus half an apple. After the third gate she has no apples left. How many had she originally ? The SolutionThe trick here is to start at the end -     1. At the third gate she gave "half of her apples plus half an apple"          and had none left.               "Half of her apples" was evidently also half an apple                    and she had one whole apple on arriving at the gate.                         NOTE that no knife was used, the last apple remained intact.     2. At the second gate she also gave "half of her apples plus half an apple"          and had one left.               "Half of her apples" was evidently an apple and a half                    and she had three apples on arriving at the gate.                         Still no knife used.     3. At the first gate she also gave "half of her apples plus half an apple"          and had three left.               "Half of her apples" was evidently three and a half apples                    and she had seven apples on arriving at the gate.                         Still no knife used. MathematicsIf "n" represents the number of gates,     the number of apples at the very start will be 2n - 1.               This is a very common formula, applying to a considerable number of games.          For 10 gates there were 1023 apples.