A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality
Clifford A Pickover
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sequence, which has dozens of fascinat- eral formula for the n th number in the ing properties. There are many ways to sequence is (1/2) × n × (3 n – 1). The first few generate the Morse-Thue sequence. One way are 1, 5, 12, 22, 35, 51, 70, and 92. Curiously, is to visualize this as a sequence of 0s and 1s. all numbers of such a type end in 0, 1, 2, 5, 6, Start with a zero and then repeatedly do the or 7. This problem can also be solved simply following replacements: 0 → 01 and 1 → 10.
5,230,096,303,003,196,309,630,967 give this particular example. He did give other to impressive examples of almost integers, such as eπ 58 . 5217 These digits sum to 15, which is divisible by 2.79 Cube square puzzle. N is 69. So 692 = 3. Thus 5,230,096,303,003,196,309,630,967 4,761 and 693 = 328,509. I believe there may cannot be a prime number. be various observations that we can make to make this problem easier to solve. For exam- 2.77 Never prime? No. Here is a counter- ple, it
container, an n = 9 hyperlattice in the hypersphere without any intersections is 50th dimension can hold each electron, each sometimes called the Newton number or proton, and each neutron in the universe (each ligancy. Newton correctly believed that the particle in its own cage). Does this all boggle kissing number in three dimensions was 12. your mind? Answers 355 4.79 In the garden of the knight. Here is the these kinds of problems (Ed Pegg Jr., “Math solution (Ed Pegg Jr., “Math
at minus 40 degrees,” 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, . . . he tells his assistant, Boris. Boris replies, “Is that Notice how 8 and 9 are consecutive integers. Are there any Centigrade or Fahrenheit?” other consecutive numbers in this sequence? For years, mathe- With his dark eyes, the sci- maticians and computer scientists have searched for other entist looks at Boris and says, examples, but, besides 8 and 9, no consecutive powers were “It doesn’t matter.” Why did ever
antimagic square. (See Answer 4.72.) Alphamagic Square Resulting Square where the magic square on the right corresponds to the num- 8 9 ber of letters in 12 15 five twenty-two eighteen 1 14 twenty-eight fifteen two 15 4 twelve eight twenty-five Antimagic Square Numbers Spelled Out In other words, you can spell out the numbers in the first Annihilation magic squares. magic square. Then count the letters in the words. The integers According to Ivan make a second magic