F IR S T A N S W E R S
of balls in a tetrahedral packing. In other words, find a number that is both tetrahedral and square. Two trivial solutions are 1 and 4. There is only one other. Can you find it?
The Supreme Ruler’s plan will not work. Consider all first-born children. One-third will be male, one-third female, one-third bisexual. Mothers who give birth to bisexuals will be sterilized. The remaining mothers may have second children. One-third of the second-bom will be male, one-third female, one-third bisexual. Again, mothers of the bisexuals will be made sterile. The remaining mothers may have third children, and so on. This obviously generalizes to families of any size. The proportions of sexes will always be 1:1:1. Assume that the decree lasts a thousand years and that all mothers live long enough, and are healthy enough, to keep bearing children until they have a bisexual. What will be the average number of children born to a Byronian mother during the millennial period?
The algorithm uses the binary system. Take 1 doyle from the first can, 2 from the second, 4 from the third, 8 from the fourth, 16 from the fifth, and 32 from the sixth. These num bers, 1, 2, 4, 8, . . . , are powers of 2, and every integer is the sum of a unique set of such powers, provided no two are alike. Place the 63 doyles on the scale and write down the excess weight in milligrams as a binary number. The posi tion of each 1 in the number, counting from the right, iden
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