Definately one of my all time favorites. Mackay's book also contains a brilliant solution to the weighing problem due to Conway, where you label the 12 spheres

Definately one of my all time favorites. Mackay's book also contains a brilliant solution to the weighing problem due to Conway, where you label the 12 spheres and the sequence of weighings is always the same (meaning that the spheres you choose to weigh on the second or third weighing don't depend on the outcome of any previous weighing. After three weighings, the outcomes of the weighings identifies the odd ball and whether it is heavier or lighter by a lookup table, which is cleverly constructed through a symmetry argument. It completely blew my mind.