Turn, Turn, Turn - Futility Closet

New Jersey magician Karl Fulves invented this ESP trick. Hand a friend an ordinary die and turn your back. Ask her to place the die on a table. Now ask her to give the die a quarter turn: If the top number is even, she must turn it to the east (to her right), and if the top number is odd, she must turn it north (away from her). This exposes a new top number, and she can turn the die again according to the same rule, turning it east if the number is even and north if it’s odd. After she has continued in this way for several turns, you ask her to stop when the top number is 1, then to give the die one final turn and to concentrate on the top number. It would seem as though the final number might be any one of four possibilities, but you can name it correctly with your back turned. How?

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After at most three moves, the die enters a recurring loop: 1 4 5 6 3 2. So after the first few turns, it’s safe to predict that 4 will follow 1. Because the whole cycle is predictable, the trick can be repeated a few times to produce various “final” numbers.