Professor Stewart's Hoard of Mathematical Treasures (4 page)

... 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

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

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:
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

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 gemowe
³
lines of one lengthe: ========, bicause noe .2. thynges, can be moare equalle.’

Robert Recorde and his equals sign.

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, impressed beyond words, caved in.

How did she do that?

Can a similar method make a six-pointed star?

 

Answers on page 278