Forty Cubed, Forty Squared, and the Number You Never Choose
A safe takes three numbers from a dial marked 1 to 40, so there are 64,000 combinations, and the worst case is 1600 attempts rather than 64,000. The third number is supplied by the mechanism instead of guessed, which collapses the search from three dimensions to two, and 1600 is proved both achievable and unavoidable. A dial with a mark of mechanical slack drops the count to 196, which is a covering problem on a cycle of forty.