Professor Stewart's Hoard of Mathematical Treasures (3 page)

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.

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:

‘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.

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.

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 ...