Mathematical problems mostly come in two kinds. There are the old and easy ones, like “What do you get when you multiply six by seven?,” whose answers can only excite seven-year-olds and autistic adults. Then there are the problems that interest research mathematicians, whose incomprehensibility begins to fade only after several university degrees in the higher algebra. It is hard to make dinner-party conversation or requests for funding out of a discipline whose very terms are well beyond the reach of ordinary mortals. There are a few mathematical questions that can be understood by anyone but whose solutions have been achieved only by the last few decades of difficult research. John Casti’s book presents five of them.

Sir Walter Raleigh asked his mathematical adviser for a formula for the number of cannonballs in various stacks on the deck of his ship. The adviser came up with something, but realized it was not clear whether the balls were being stacked in the most efficient way. There is a natural way to pack spheres tightly: lay down the deck layer in “hexagonal” fashion, where each ball is surrounded by six others. Then put the second layer so that the balls lie in every second depression in the first layer; the second layer will then also form a hexagonal packing. Then put the third layer in every second depression of the second layer, so that the balls of the third layer are directly above those of the first. If this is