Ian Stewart Read online

  Which year in the past is the most recent one that reads the same when you turn it upside down?

  Which year in the future is the next one that reads the same when you turn it upside down?

  Answers on page 274

  Luckless Lovelorn Lilavati

  Lilavati.

  Among the great mathematicians of ancient India was Bhaskara, ‘The Teacher’, who was born in 1114. He was really an astronomer: in his culture, mathematics was mainly an astronomical technique. Mathematics appeared in astronomy texts; it was not a separate subject. Among Bhaskara’s most famous works is one named Lilavati. And thereby hangs a tale.

  Fyzi, Court Poet to the Mogul Emperor Akbar, wrote that Lilavati was Bhaskara’s daughter. She was of marriageable age, so Bhaskara cast her horoscope to work out the most propitious wedding date. (Right into the Renaissance period, many mathematicians made a good living casting horoscopes.) Bhaskara, clearly a bit of a showman, thought he’d come up with a terrific idea to make his forecast more dramatic. He bored a hole in a cup and floated it in a bowl of water, with everything designed so that the cup would sink when the fateful moment arrived.

  Unfortunately, an eager Lilavati was leaning over the bowl, hoping that the cup would sink. A pearl from her dress fell into the cup and blocked the hole. So the cup didn’t sink, and poor Lilavati could never get married.

  To cheer his daughter up, Bhaskara wrote a mathematics textbook for her.

  Hey, thanks, Dad.

  Sixteen Matches

  Sixteen matches are arranged to form five identical squares.

  By moving exactly two matches, reduce the number of squares to four. All matches must be used, and every match should be part of one of the squares.

  Answer on page 274

  Sixteen matches arranged to form five squares.

  Swallowing Elephants

  Elephants always wear pink trousers.

  Every creature that eats honey can play the bagpipes.

  Anything that is easy to swallow eats honey.

  No creature that wears pink trousers can play the bagpipes.

  Therefore:

  Elephants are easy to swallow.

  Is the deduction correct, or not?

  Answer on page 274

  Magic Circle

  In the diagram there are three big circles, and each passes through four small circles. Place the numbers 1, 2, 3, 4, 5, 6 in the small circles so that the numbers on each big circle add up to 14.

  Answer on page 276

  Make the sum 14 around each big circle.

  Dodgem

  This is a mathematical game with very simple rules that’s a lot of fun to play, even on a small board. It was invented by puzzle expert and writer Colin Vout. The picture shows the 4×4 case.

  Dodgem on a 4×4 board.

  Players take turns to move one of their counters one cell forward, one to the left, or one to the right, as shown by the arrows ‘black’s directions’ and ‘white’s directions’. They can’t move a counter if it is blocked by an opponent’s counter or the edge of the board, except for the opposite edge where their counters can escape. A player must always leave his opponent at least one legal move, and loses the game if he does not. The first player to make all his counters escape wins.

  On a larger board the initial arrangement is similar, with the lower left-hand corner unoccupied, a row of white counters up the left-hand column, and a row of black counters along the bottom row.

  Vout proved that with perfect strategy the first player always wins on a 3×3 board, but for larger boards it seems not to be known who should win. A good way to play is with draughts (checkers) pieces on the usual 8×8 board.

  It seems natural to use square boards, but with a rectangular board the player with fewer counters has to move them further, so the game may be playable on rectangular boards. As far as I know, they’ve not been considered.

  Press-the-Digit-ation

  I learned this trick from the Great Whodunni, a hitherto obscure stage magician who deserves wider recognition. It’s great for parties, and only the mathematicians present will guess how it works.1 It is designed specifically to be used in the year 2009, but I’ll explain how to change it for 2010, and the Answer section on page 276 will extend that to any year.

  Whodunni asks for a volunteer from the audience, and his beautiful assistant Grumpelina hands them a calculator. Whodunni then makes a big fuss about it having once been a perfectly ordinary calculator, until it was touched by magic. Now, it can reveal your hidden secrets.

  ‘I am going to ask you to do some sums,’ he tells them. ‘My magic calculator will use the results to display your age and the number of your house.’ Then he tells them to perform the following calculations:

  • Enter your house number.

  • Double it.

  • Add 42.

  • Multiply by 50.

  • Subtract the year of your birth.

  • Subtract 50.

  • Add the number of birthdays you have had so far this year, i.e. 0 or 1.

  • Subtract 42.

  ‘I now predict,’ says Whodunni, ‘that the last two digits of the result will be your age, and the remaining digits will be the number of your house.’

  Let’s try it out on the fair Grumpelina, who lives in house number 327. She was born on 31 December 1979, and let’s suppose that Whodunni performs his trick on Christmas Day 2009, when she is 29.

  • Enter your house number: 327

  • Double it: 654.

  • Add 42: 696.

  • Multiply by 50: 34,800.

  • Subtract the year of your birth: 32,821

  • Subtract 50: 32,771.

  • Add the number of birthdays you have had so far this year (0): 32,771.

  • Subtract 42: 32,729.

  The last two digits are 29, Grumpelina’s age. The others are 327—her house number.

  The trick works for anyone aged 1 to 99, and any house number, however large. You could ask for a phone number instead, and it would still work. But Grumpelina’s phone number is ex-directory, so I can’t illustrate the trick with that.

  If you’re trying the trick in 2010, replace the last step by ‘subtract 41’.

  You don’t need a magic calculator, of course: an ordinary one works fine. And you don’t need to understand how the trick works to amaze your friends. But for those who’d like to know the secret, I’ve explained it on page 276.

  Secrets of the Abacus

  In these days of electronic calculators, the device known as an abacus seems rather outmoded. Most of us encounter it as a child’s educational toy, an array of wires with beads that slide up and down to represent numbers. However, there’s more to the abacus than that, and such devices are still widely used, mainly in Asia and Africa. For a history, see: en.wikipedia.org/wiki/Abacus

  The basic principle of the abacus is that the number of beads on each wire represents one digit in a sum, and the basic operations of arithmetic can be performed by moving the beads in the right way. A skilled operator can add numbers together as fast as someone using a calculator can type them in, and more complicated things like multiplication are entirely practical.

  The Sumerians used a form of abacus around 2500 BC, and the Babylonians probably did too. There is some evidence of the abacus in ancient Egypt, but no images of one have yet been found, only discs that might have been used as counters. The abacus was widely used in the Persian, Greek and Roman civilisations. For a long time, the most efficient design was the one used by the Chinese from the 14th century onwards, called a suànpán. It has two rows of beads; those in the lower row signify 1 and those in the upper row signify 5. The beads nearest the dividing line determine the number. The suànpán was quite big: about 20 cm (8 inches) high and of varying width depending on the number of columns. It was used lying flat on a table to stop the beads sliding into unwanted positions.

  The number 654,321 on a Chinese abacus.

  The Japanese imported
the Chinese abacus from 1600, improved it to make it smaller and easier to use, and called it the soroban. The main differences were that the beads were hexagonal in cross-section, everything was just the right size for fingers to fit, and the abacus was used lying flat. Around 1850 the number of beads in the top row was reduced to one, and around 1930 the number in the bottom row was reduced to four.

  Japanese abacus, cleared.

  The first step in any calculation is to clear the abacus, so that it represents 0 . . . 0. To do this efficiently, tilt the top edge up so that all the beads slide down. Then lie the abacus flat on the table, and run your finger quickly along from left to right, just above the dividing line, pushing all the top beads up.

  Japanese abacus, representing 9,876,543,210.

  Again, numbers in the lower row signify 1 and those in the upper row signify 5. The Japanese designer made the abacus more efficient by removing superfluous beads that provided no new information.

  The operator uses the soroban by resting the tips of the thumb and index finger lightly on the beads, one either side of the central bar, with the hand hovering over the bottom rows of beads. Various ‘moves’ must then be learned, and practised, much like a musician learns to play an instrument. These moves are the basic components of an arithmetical calculation, and the calculation itself is rather like playing a short ‘tune’. You can find lots of detailed abacus techniques at: www.webhome.idirect.com/~totton/abacus/pages.htm#Soroban1

  I’ll mention only the two easiest ones.

  A basic rule is: always work from left to right. This is the opposite of what we teach in school arithmetic, where the calculation proceeds from the units to the tens, the hundreds, and so on - right to left. But we say the digits in the left-right order: ‘three hundred and twenty-one’. It makes good sense to think of them that way, and to calculate that way. The beads act as a memory, too, so that you don’t get confused by where to put the ‘carry digits’.

  To add 572 and 142, for instance, follow the instructions in the pictures. (I’ve referred to the columns as 1, 2, 3, from the right, because that’s the way we think. The fourth column doesn’t play any role, but it would do if we were adding, say, 572 and 842, where 5 + 8 = 13 involves a ‘carry digit’ 1 in place 4.

  Set up 572

  Add 1 in column 3

  Add 4 in column 2 ...

  ... and carry the 1

  Add 2 in column 3

  A basic technique occurs in subtraction. I won’t draw where the beads go, but the principle is this. To subtract 142 from 572, change each digit x in 142 to its complement 10 - x. So 142 becomes 968. Now add 968 to 572, as before. The result is 1,540, but of course 572 - 142 is actually 430. Ah, but I haven’t yet mentioned that at each step you subtract 1 from the column one place to the left (doing this as you proceed). So the initial 1 disappears, 5 turns into 4, and 4 turns into 3. The 0 you leave alone.

  Why does this work, and why do we leave the units digit unchanged?

  Answer on page 277

  Redbeard’s Treasure

  Captain ‘Jolly’ Roger Redbeard, the fiercest pirate in the Windlass Islands, stared blankly at a diagram he had drawn in the sand beside the quiet lagoon behind Rope’s End Reef. He had buried a hoard of pieces of eight there a few years ago, and now he wanted to retrieve his treasure. But he had forgotten where it was. Fortunately he had set up a cunning mnemonic, to remind him. Unfortunately, it was a bit too cunning.

  He addressed the band of tattered thugs that constituted his crew.

  ‘Avast, ye stinkin’ bilge-rats! Oi, Numbskull, put down that cask o’ rotgut and listen!’

  The crew eventually quietened down.

  ‘You remember when we boarded the Spanish Prince? And just before I fed the prisoners to the sharks, one of ’em told us where they’d hidden their loot? An’ we dug it all up and reburied it somewhere safe?’

  There was a ragged cry, mostly of agreement.

  ‘Well, the treasure is buried due north o’ that skull-shaped rock over there. All we need to know is how far north. Now, I’appens to know that the exact number o’ paces is the number of different ways a man can spell out the word TREASURE by puttin’ his finger on the T at the top o’ this diagram, and then movin’ it down one row at a time to a letter that’s next to it, one step to the left or right.

  ‘I’m offerin’ ten gold doubloons to the first man-jack o’ ye to tell me that number. What say ye, lads?’

  How many paces is it from the rock to the treasure?

  Answer on page 277

  Hexaflexagons

  These are fascinating mathematical toys, originally invented by the prominent mathematician Arthur Stone when he was a graduate student. I’ll show you the simplest one, and refer you to the internet for the others.

  Cut out a strip of 10 equilateral triangles and fold the right-hand end underneath the rest along the solid line . . .

  . . . to get this. Now fold the end backwards along the solid line and poke it through . . .

  . . . to get this. Finally, fold the grey flap behind and glue it to the adjacent triangle . . .

  . . . to get a finished triflexagon.

  Having made this curious shape, you can flex it. If you pinch together two adjacent triangles separated by a solid line (the edge of the original strip), then a gap opens in the middle and you can flip the edges outwards - turning the hexagon inside out, so to speak. This exposes a different set of faces. It can then be flexed again, which returns it to its starting configuration.

  How to flex your hexaflexagon.

  All this is easier to do by experimenting on a model than to describe. If you colour the front of the original hexagon red, and the back blue, then the first flex reveals another set of triangles that have not yet been coloured. Colour these yellow. Now each successive flex sends the front colour to the back, makes the back colour disappear, and shows a new colour on the front. So the colours cycle like this:

  • Red on front, blue on back

  • Yellow on front, red on back

  • Blue on front, yellow on back

  There are more complicated flexagons, with more hidden faces, which require more colours. Some use squares instead of triangles. Stone formed a ‘flexagon committee’ with three other graduate students: Richard Feynman, Brent Tuckerman and John Tukey. In 1940, Feynman and Tukey developed a complete mathematical theory characterising all flexagons. A good entry point into the extensive world of the flexagon is en.wikipedia.org/wiki/Flexagon

  Who Invented the Equals Sign?

  The origins of most mathematical symbols are lost in the mists of antiquity, but we do know where the equals sign = came from. Robert Recorde was a Welsh doctor and mathematician, and in 1557 he wrote The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.2

  In it, he wrote: ‘To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe3 lines of one lengthe: ========, bicause noe .2. thynges, can be moare equalle.’

  Robert Recorde and his equals sign.

  Stars and Snips

  Betsy Ross, who was born in 1752, is generally credited with having sewn the first American flag, with 13 stars representing the 13 founding colonies. (On the present-day Stars and Stripes, they are represented by the 13 stripes.) Historians continue to debate the truth of this story, since it is mainly based on word of mouth, and I don’t want to get tangled up in the historical arguments: see www.ushistory.org/betsy/

  The important thing for this puzzle is that the stars on the American flag are five-pointed. Apparently George Washington’s original design used six-pointed stars, whereas Betsy favoured the five-pointed kind. The committee objected that this type of star was too hard to make. Betsy picked up a piece of paper, folded it, and cut off a perfect five-pointed star with one straight snip of her scissors. The committee, impre
ssed beyond words, caved in.

  How did she do that?

  Can a similar method make a six-pointed star?

  Answers on page 278

  Fold and cut this . . .

  . . . to make this.

  By the Numbers of Babylon

  Ancient cultures wrote numbers in many different ways. The ancient Romans, for instance, used letters: I for 1, V for 5, X for 10, C for 100, and so on. In this kind of system, the bigger the numbers become, the more letters you need. And arithmetic can be tricky: try multiplying MCCXIV by CCCIX, using only pencil and paper.

  Our familiar decimal notation is more versatile and better suited to calculation. Instead of inventing new symbols for ever-bigger numbers, it uses a fixed set of symbols, which in Western cultures are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Larger numbers are taken care of by using the same symbols in different positions. For instance, 525 means