Professor Stewart’s Superlative Sneaky Solutions // 323
simplest, but there are infinitely many others, even if you restrict the entries to positive numbers.
An unorthodox magic square.
........................................... The Look and Say Sequence The rule of formation is best stated in words. The first term is ‘1’, which can be read as ‘one one’, so the next term is 11. This reads ‘two ones’, leading to 21. Read this as ‘one two, one one’ and you see where 1211 comes from, and so on. Conway proved that if L(n) is the length of the nth term in this sequence, then LðnÞ&ð1:30357726903. . .Þn where 1.30357726903 . . . is the smallest real solution of the 71stdegree polynomial equation x71 x69 2x68 x67 þ 2x66 þ 2x65 x63 x62 x61 x60 þ 2x58 þ 5x57 þ 3x56 2x55 10x54 3x53 2x52 þ 6x51 þ 6x50 þ x49 þ 9x48 3x47 7x46 8x45 8x44 þ 10x43 þ 6x42 þ 8x41 5x40 12x39 þ 7x38 7x37 þ 7x36 þ x35 3x34 þ 10x33 þ x32 6x31 2x30 10x29 3x28 þ 2x27 þ 9x26 3x25 þ 14x24 8x23 7x21 þ 9x20 þ 3x19 4x18 10x17 7x16 þ 12x15 þ 7x14 þ 2x13 12x12 4x11 2x10 þ 5x9 þ x7 7x6 þ 7x5 4x4 þ 12x3 6x2 þ 3x 6 ¼0
........................................... I don’t claim this is obvious.