CLASSIC BRAINTEASERS AND OTHER TIMELESS MATHEMATICAL GAMES OF THE LAST 10 CENTURIES DOMINIC OLIVASTRO [...]... F DAYS WERE BOTH I N S T A N C E S O F T H E NUMBER TWO —BERTRAND RUSSELL H OW ANCIENT IS THE MOST ANCIENT PUZZLE? It is a fairly simple matter to find an ancient manuscript recounting the popular puzzles of its time, but such manuscripts will take us back to the second millennium B.C at the earliest Surely, the greatest puzzles of all must be those that were never recorded, the ones that were invented... distant place Although this book contains a few (very interesting) superstitions, I hope it will be taken the right way It is meant to flesh out ancient puzzles; it is not meant to support modern foolishness My attempt in each chapter is to begin with ancient puzzles and move as quickly as possible to more modern problems that suggest themselves One could write several volumes this way, but by necessity... it mentioned often elsewhere, and it gives me the opportunity to bring in puzzles of indeterminate equations There are many glaring omissions, and the one of which I am most ashamed is the complete absence of Native Americans Since the chapters are arranged in a roughly chronological order, the book as a whole follows a similar ancient- to-modern design It begins with a bone, and it ends with a tree 3... barricade and come together can produce offspring This, too, is nonsense, but now it is dull and childish nonsense.3 Unlike ancient number mysticism, it does not lead to new insights and it will never disappear It is said by a people who have grown disenchanted with the world Ancient supersitions were always forwardlooking Modern irrationalities look backward The apricot pit that cures cancer, the... bone Next, consider the way the carvers of the bone were mystified by doubling a number This is another common feature of ancient mathematics, found in many regions of Africa and elsewhere An extended use E T C H E S 11 12 A N C I E N T P U Z Z L E S of doubling, certainly of very ancient origins, is found in modern Ethiopia The story is told of a colonel who wished to purchase seven bulls, each costing... this, however, because all we want to show is how close the function R(n) comes to predicting the number of primes less than or equal to n We do this in Figure 5 Figure 4 The distribution of primes 18 ANCIENT PUZZLES (2) (3) (4) Number of primes less than or equal to the power of ten R(n) Difference (1) Powers of ten 100,000,000 200,000,000 300,000,000 400,000,000 500,000,000 600,000,000 700,000,000 800,000,000... 13/11 12: 910: 13: 170 14: 156 10 7 I 7/5 8 8 11/13 15: 132 16: 116 17: 308 I 2Д I 9З I 13/11 18: 19: Done: 364—, 68 I 1/1 7 9 < 8 ГЙП 4=2J Figure 8 Finding the first prime with the Sieve of Conway 22 ANCIENT PUZZLES How in the world does it work? To answer that, let us look at something that may seem unrelated at first but which is really very similar to the Conway sieve Imagine a simple computer with... machine is currently in state 5 and register A is not equal to 0; otherwise the fraction will not produce a whole number What happens ETCHES 23 Figure 9 How an Ishango computer subtracts two numbers 24 ANCIENT PUZZLES Figure 10 The meaning of the fractions 0: Load Register A and Register В 1: Is B=0? If yes, go to step 5 2: B = B-1 Г 3: A = A-l • Г 4: Go to step 1 3x5 2x5_ 7 1 5: Stop 5 5 (A) (B) (C) (D)... (C) 13 7 (D) 17 65 (E) 39 34 (F) 7 13 (G) 19 17 (H) 23 57 (I) 38 23 (J) 1 19 (K) 1 5 (L) 7 1 (b) Conway fractions that divide В by A Figure 11 How an Ishango computer divides two numbers Stop 25 26 ANCIENT PUZZLES Use the computer—use the fractions—to see if 3 is divisible by 2 This time we need three registers, since a third is used for temporary storage We load the numbers into registers A and В respectively,... once for two sheep The transaction had to be conducted more slowly—first two sticks of tobacco were given for one sheep, then two sticks were given again for a second sheep When it E T C H E S 29 30 ANCIENT PUZZLES was pointed out that the final transaction was the same as the one originally offered, the Damaras were suspicious of the trader, as though he possessed magical powers There are many stories . way. It is meant to flesh out ancient puzzles; it is not meant to support modern foolishness. My attempt in each chapter is to begin with ancient puzzles and move as quickly . NUMBER TWO. —BERTRAND RUSSELL H OW ANCIENT IS THE MOST ANCIENT PUZZLE? It is a fairly simple matter to find an ancient manuscript recounting the popular puzzles of its time, but such manu- scripts. another common feature of ancient mathe- matics, found in many regions of Africa and elsewhere. An extended use 12 ANCIENT PUZZLES of doubling, certainly of very ancient origins, is found