by Alex Bellos
Before you look in the back, have a go at sketching what you think you might see, but don’t feel bad if you’re finding it hard. To deduce the pattern requires phenomenal levels of spatial intuition. I’ve included the problem not to show you up but to give you a thrill. The geometer who showed me this problem said that seeing the solution was the biggest wow he has ever experienced in mathematics.
Prince Rupert was concerned with pushing a cube through a square hole. Now we’re concerned not just with square holes but with circular and triangular ones too.
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THE PECULIAR PEG
Draw a three-dimensional picture of a solid object that will pass through square, circular and triangular holes, as illustrated above. The object must touch every point on the inside of each hole as it passes through. In other words, the object has three cross-sections that are the same shape and size as these three holes.
Those readers who already know of one solution to this problem, should draw a different object. Or does only one object fit the bill?
My final problem about visualisation in three dimensions flummoxed a panel of more than a dozen university professors in America. They approved it for use in an aptitude test given in 1980 to 1.3 million secondary school pupils.
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THE TWO PYRAMIDS
The faces of the two pyramids shown above are equilateral triangles of the same size. One of the pyramids has a triangular base and the other a square base. If the pyramids are glued together so that two triangular faces perfectly coincide, how many exposed faces would the resulting solid have?
(a) 5 (b) 6 (c) 7 (d) 8 (e) 9
The answer given by the examination board was (c), the wrong answer. The board assumed that if one pyramid has 4 sides and the other 5 sides, when you stick them together they each lose a side. Actually, that’s not the case. The error was only picked up when one of the students who took the test, Daniel Lowen, aged 17, got his results. After the original exam, Lowen had made a physical model of the two pyramids, which confirmed the answer he had written down. However, when the results were released he discovered he’d got that question wrong. His father – an engineer working on the space shuttle; yes, a rocket scientist! – tried to show his son why his answer was wrong, but ended up proving that it was right. The father contacted the examination board, which apologised, and the story later made the front page of the New York Times. You might be able to guess the correct answer without any help, but if you want a tip, you might find it useful to place two square pyramids together, as shown below.
The next question was also set in an exam. In 1995, the inaugural Trends in International Mathematics and Science Study – TIMSS – became the largest international assessment of the abilities of students around the world, testing students across 41 education systems. The following question was given to 18-year-olds in 16 countries studying ‘advanced’ maths, meaning A-level or similar.
Overall, only 10 per cent of students got the right answer. Top of the class was Sweden, where 24 per cent got it correct. In the US and France, the score was a measly 4 per cent. The UK did not take part.
I’m including the problem here because it has a ‘simple’ solution, and requires no technical maths beyond what should be known by a 14-year-old. Sometimes knowing too much maths can be a disadvantage.
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THE ROD AND THE STRING
A string is wound symmetrically around a circular rod. The string goes exactly four times around the rod. The circumference of the rod is 4cm and its length is 12cm. Find the length of the string.
Just as the circle is the simplest two-dimensional shape, the sphere is the simplest three-dimensional object. The Earth is roughly spherical. In fact, for the next problem we can assume that the Earth is a perfect sphere.
A well-known puzzle concerns a person who walks one hundred miles due south, one hundred miles due west, and finally one hundred miles due north, only to return to the very point where they started. The question asks: What colour is the bear?
White, of course. The only bear species that lives in a region where this three-legged trek is possible is the polar bear. If the traveller starts at the North Pole, a trip one hundred miles due south, one hundred miles due west and one hundred miles due north traces a triangle that returns them to their point of departure, the North Pole. Which leads us to the following shaggy tale.
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WHAT COLOUR IS THE BEARD?
A person who walks 10 miles due north, then 10 miles due west, and then 10 miles due south, finds themselves back where they started. They are not at the South Pole.
What colour is the beard?
The remaining compass direction is east.
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AROUND THE WORLD IN 18 DAYS
In an updated version of Jules Verne’s novel Around the World in Eighty Days, Phileas Fogg circumnavigates the globe by aeroplane. His departure point is London, and he travels through the same countries as in the original story: Egypt, India, Hong Kong, Japan and then across the US. He leaves at noon on October 2nd, and counts 18 days until he gets back. What date does he arrive in London?
To celebrate the end of this chapter, a wee dram.
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A WHISKY PROBLEM
A full whisky bottle has a height of 27cm and a diameter of 7cm, and contains 750 cubic centimetres of whisky. Like many bottles, it has a dome-like indentation at the bottom.
You have a drink, fall asleep, and wake up to discover that the bottle now contains whisky to a height of only 14cm. When you turn the bottle over, the height of whisky is 19cm.
How much whisky is still in the bottle, in cubic centimetres?
We know from a previous question that the area of a circle is πr2, where r is the radius and π is 3.14 to two decimal places. The volume of a cylinder is πr2 x h, where r is the radius and h is the height.
Try to solve the whisky problem without writing down a single equation. If you’re having trouble, you might be tempted to drink some more. Don’t pour yourself too much, though!
Tasty teasers
Pencils and utensils
1)
Put your hands together, with a pencil between your thumbs, as shown above left. Can you twist your hands so you get to the position shown above right, in which the pencil is underneath, resting between your palms and thumbs? During the twist you must not let go of the pencil.
2)
Two iron rods are on a table. One is magnetised, with a pole at each end. How can you tell which one is the magnet if all you are allowed to do is move them around the table with your fingers, without using any other instrument or lifting them up?
3)
A dentist has three patients to see, but only two pairs of sterile gloves. How can the dentist attend to all three patients (wearing gloves) without the risk of transferring germs from any one person to another person via contaminated gloves? You can assume she has an assistant on hand to help her take the gloves on and off.
4)
There’s a classic puzzle that asks you to make a square with two forks on a rectangular table. The solution is to place the two forks perpendicular to each other at one corner of the table; using two sides of the table you create the square. But can you take the same two forks and make four squares instead?
5) How can you cut a hole in a postcard that is big enough for a person to get through?
A wry plod
WORD PROBLEMS
I ♥ maths. I also ♥ words. In fact, it is common for maths-loving folk to display a partiality to words, puns and wordplay. Maths is all about playing with combinations. Wordplay just swaps abstract symbols for symbols in word form. This part of my book is about having fun with our ABCs. You will confront conundrums about writing constraints, translation, punctuation, fonts and anagrams. (Look up! Can you crack ‘A wry plod’?) My initial task for you, though, right now, is to find what is so linguistically atypical about this paragraph. An unusual thing is going on, without a doubt. If you look particu
larly scrupulously at what is in front of you, it will dawn on you fairly soon. A tip: go through this paragraph word by word. Do it now, prior to carrying on.
I hope you solved it with ease. I mean, without ‘e’s. The previous paragraph did not contain the most common letter in the English language, the alphabetical character known to its detractors as the filthy fifth glyph.
A piece of text that omits a particular letter, or set of letters, is called a lipogram and is an example of ‘constrained writing’, a technique in which writers enforce some kind of pattern on their writing, or follow a rule that forbids certain things. The earliest known lipogram is a Greek poem from the sixth century BCE, written deliberately without the letter sigma. The genre’s most impressive achievement, however, is the French writer Georges Perec’s novel La Disparition, from 1969. This 300-page book contains no occurrences of the letter ‘e’, which is even more common in French than it is in English. A good lipogram disguises its constraint. A sign of Perec’s virtuosity in composing fluent ‘e’-less text is that when La Disparition came out, some readers did not notice that the letter was missing, even though its absence was a central theme in the plot. (The literal translation of La Disparition is The Disappearance, but since these words include three ‘e’s, Gilbert Adair titled his 1994 English translation of the book, A Void.)
Many writers are drawn to literary constraints. Abiding by a strict rule can be unexpectedly liberating, and produce unusual, ingenious and beautiful results. Indeed, Perec was a member of a collective of writers and mathematicians, Oulipo, the Ouvroir de Littérature Potentielle, or the Workshop of Potential Literature, which over the past 50 years has created many literary works based on mathematical rules. In the spirit of Oulipo, the puzzles in this chapter also celebrate the playful interplay between mathematics and language. Word up.
The inverse challenge to writing text without an ‘e’ is to write text in which the only vowel allowed is an ‘e’. Such as Persevere ye perfect men, Ever keep these precepts ten, instructions for abiding by the Ten Commandments.
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THE SACRED VOWELS
Below are five sentences with the vowels and spaces taken out. Your task is to reinsert the vowels to recreate the sentences. Each sentence uses one vowel only. The five vowels – A, E, I O and U – each have their own sentence.
[1] L L N T H J Z B L C H W S S W T P R S R V S T H N B L C H S B R
[2] R T H D X M N K T T W L F S D W N T W B W L S F P R K W N T N
[3] N R B L S S G N W S T L M B S H N K N D H S J M T R T S S S N C K
[4] B D S D R N K C H M G L G S R M P N C H, P C H C K S H M M S B R N C H, S L M P S
[5] T H S R C H D S H S F G S N C N G, W H C H F N S H W H L S T S W G G N G G N
Each sentence involves food and drink.
If you find univocalic text enthralling, as I do, I strongly recommend Eunoia, from 2001, by the Canadian writer Christian Bök. A masterpiece of constrained writing, it has five chapters, each of which allows only a single vowel. The previous problem was inspired by that book.
In fact, as a subsidiary challenge, choose a vowel and try to write a meaningful sentence of at least seven words that uses only that vowel. You will find ‘a’ and ‘e’ sentences the easiest, ‘i’ and ‘o’ more difficult, and ‘u’ the hardest. U have been warned.
Mary Youngquist was the first woman to get a PhD in organic chemistry from MIT. A fan of puzzles, she was editor of the US National Puzzler’s League newsletter for six years in the 1970s. She also wrote the following poem, which hides a very simple constraint.
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WINTER REIGNS
Shimmering, gleaming, glistening glow
Winter reigns, splendiferous snow!
Won’t this sight, this stainless scene,
Endlessly yield days supreme?
Eying ground, deep piled, delights
Skiers scaling garish heights.
Still like eagles soaring, glide
Eager racers; show-offs slide.
Ecstatic children, noses scarved
Dancing gnomes, seem magic carved
Doing graceful leaps. Snowballs,
Swishing globules, sail low walls.
Surely year-end’s special lure
Eases sorrow we endure,
Every year renews shared dream,
Memories sweet, that timeless stream.
What simple rule governs this verse?
The poem may have little literary merit, a comment I’m sure Youngquist would have agreed with. Its charm and inventiveness lie in the way that despite such a stringent constraint, it reads like a poem, and makes sense.
Almost all of Oulipo’s output is in French. Two modern champions of constrained writing in English are the Americans Mike Keith and Doug Nufer. The former is best known for his work in ‘Pilish’, a rule that states that word lengths must follow the digits in pi, the number whose first six decimal digits are 3.14159 and whose remaining digits continue ad infinitum without ever repeating a pattern. In a piece of Pilish text, the first word has 3 letters, the second 1 letter, the third 4, and so on. Keith’s book, Not A Wake, follows the first 10,000 digits of pi. It begins: Now I fall, a tired suburbian…
One mathematical pattern evident in almost all text – books, articles, emails – is that about half the words used appear more than once. Doug Nufer’s magnum opus is a 40,000-word novel, Never Again, in which every word appears exactly once. (He counts plurals as different words). To break the universal law of word frequency while still saying something meaningful at such length is a remarkable achievement. His book starts: When the racetrack closed forever I had to get a job. And ends: Worldly bookmaker soulmates rectify unfair circumstance’s recurred tragedies, ever-moving, ever-hedging shifty playabilities since chances say someone will be for ever closing racetracks.
If you try to write a paragraph in Pilish, or one with no repeated words, you will quickly appreciate quite how impressive Keith and Nufer’s work is.
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FIVE DEFT SENTENCES
Each of the sentences below is written according to a different constraint. Can you work out what each constraint is?
[1] Deft Afghans hijack somnolent understudies.
[2] Dennis, Nell, Edna, Leon, Anita, Rolf, Nora, Alice, Carol, Lora, Cecil, Aaron, Flora, Tina, Noel and Ellen sinned.
[3] Quiet Pete wrote poor poetry. Had alfalfa dhal (half a flask). Mmmmm.
[4] I do not know where family doctors acquired illegibly perplexing handwriting.
[5] a wise unicorn swims in a ravine, saves an anaemic racoon.
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THE CONSONANT GARDENER
Plant a consonant in each cell to make five words
The word at the top of this page is not even the longest word in English that alternates between vowels and consonants. That honour goes to ‘honorificabilitudinitatibus’, which means honourableness and appears in Shakespeare’s Love’s Labour’s Lost.
A kangaroo carries a joey in its pouch. Likewise, kangaroo words carry smaller versions of themselves.
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KANGAROO WORDS
A kangaroo word contains the letters, in the correct order, of a synonym of itself. For example, BLOssOM includes the word ‘bloom’. For each of the kangaroo words below find the synonym it contains.
Chicken Instructor
Contaminate Separate
Deceased Myself
Fabrication Precipitation
Honourable Satisfied
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THE TEN-LETTER WORDS
Each of these words has 10 letters. What else do they have in common?
Keennesses
Nagnagging
Rememberer
Rerendered
Sleeveless
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TEN NOTABLE NUMBERS
Find the following number words.
[1] The only number word that accurately describes how many letters it has.
[2] The ten-letter number word that contains ten differ
ent letters.
[3] The largest number word without an ‘n’.
[4] The only number word with every letter in alphabetical order.
[5] The only number word with every letter in reverse alphabetical order.
[6] The smallest whole number that includes every vowel – a e i o u – in that order.
[7] The six-letter word that describes where its first letter appears in the alphabet.
[8] The lowest whole number to include an ‘a’.
[9] The lowest whole number to include a ‘b’.
[10] The lowest whole number to include a ‘c’.
In 1982, the British electronics engineer Lee Sallows devised a mind-boggling new literary constraint: the self-enumerating sentence, in which the sentence accurately lists what it contains. Here is his first attempt:
Only the fool would take trouble to verify that his sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s, four g’s, thirteen h’s, fifteen i’s, two k’s, nine l’s, four m’s, twenty-five n’s, twenty-four o’s, five p’s, sixteen r’s, forty-one s’s, thirty-seven t’s, ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single!