by Alex Bellos
In subsequent years Sallows devised many variations, such as this ‘pangram’, the name for a sentence that contains every letter of the alphabet.
This pangram contains two hundred nineteen letters: five a’s, one b, two c’s, four d’s, thirty-one e’s, eight f’s, three g’s, six h’s, fourteen i’s, one j, one k, two l’s, two m’s, twenty-six n’s, seventeen o’s, two p’s, one q, ten r’s, twenty-nine s’s, twenty-four t’s, six u’s, five v’s, nine w’s, four x’s, five y’s, and one z.
A natural conclusion to Sallows’s work on self-enumerating sentences is the self-enumerating question.
83
THE QUESTIONS THAT COUNT THEMSELVES
[1] How many letters would this question contain if the answer wasn’t already seventy-one?
[2] How many letters would the next question contain if the answer wasn’t already seventy-eight?
[3] How many letters would the previous question contain if the answer wasn’t already seventy-six?
[4] How many letters would the next question contain if the answer wasn’t already ninety-two?
[5] How many letters would these two questions jointly contain if the answer wasn’t already one hundred sixty-five?
The questions sound self-contradictory, like one-liners in a stand-up comedy routine. I’ll help you out with the first one. The question reveals that there are 71 letters in the sentence. What Sallows is asking you to do is to substitute ‘seventy-one’ with a number word that also describes the correct number of letters in the new sentence. For example, ‘seventy-two’ can’t be the answer, since ‘seventy-two’ has the same number of letters as ‘seventy-one’, meaning that the new sentence continues to have 71 letters, making the sentence incorrect. The number that works is ‘seventy-three’, since ‘seventy-three’ has two more letters than ‘seventy-one’, thus bringing the total number of letters in the sentence to 73.
The Belgian recreational mathematician Eric Angelini created the following puzzle, which also plays with words, numbers and self-reference.
84
THE SEQUENCE THAT DESCRIBES ITSELF
Here are the first four terms in a sequence of number words:
SIX, ONE, EIGHT, FIVE, …
Each of the words in the sequence describes the number of letters you must count until you reach the next E in the sequence.
SIX, ONE, / E / IGHT, FIVE / …
6 1 8
(SIX) (ONE) (EIGHT)
It takes 6 letters to get to the first E, it takes another 1 letter to get to the second E, it takes a further 8 letters to get to the third E, and so on. These numbers – 6, 1, 8, … – in words are precisely the sequence itself.
If each term in the sequence is the lowest possible number of letters to the next E, find the next six terms in the sequence.
85
SEXY LEXY
Let a number be known as lexy if the number of digits in its decimal expansion is equal to the number of letters it has when it’s described in words.
For example, ‘eleven trillion’ is lexy because ‘eleven trillion’ has 14 letters and 11,000,000,000,000 has 14 digits.
What is the lowest lexy number?
86
LETTERS IN A BOX
Find the words by placing letters in the cells. Hyphenated words are acceptable.
87
WONDERFUL WORDS
[1] From which word can you take away the whole, and yet have some left?
[2] The word ‘stressed’ is the longest English word to have which property?
[3] What links the words ‘natal’, ‘nice’ and ‘reading’?
[4] Which common English word becomes plural when an ‘a’ is added to its start?
[5] Which familiar word of five letters becomes shorter if you add two letters to it?
[6] Which English word describes the absence of a person or thing, and yet when you insert a space between two letters of this word, the two resulting words describe that person or thing as being present at this very moment?
[7] Name a five-letter word that is pronounced the same when four of its five letters are removed.
[8] Why does day begin with a ‘d’ and end with an ‘e’?
[9] In what way do these six adjectives form a sequence? Unpredictable, sexual, direct, warlike, jolly, gloomy
[10] What will always stay the same however many letters you take from it?
Here’s a simple sentence: Buffalo!
A context for this sentence might be during a country stroll when you suddenly see a herd of the animals approaching.
Here’s another sentence: Buffalo buffalo.
The verb ‘to buffalo’ is an American term meaning to intimidate, or baffle. In the context of you being intimidated, or even baffled, by some buffalo, the sentence clearly makes sense.
Another one: Buffalo buffalo buffalo.
The first word now refers to the American city of Buffalo. Again the sentence makes sense. Buffalo from the city of Buffalo intimidate.
In fact, it is possible to make grammatically correct and meaningful sentences that consist solely of the word ‘buffalo/Buffalo’, repeated as many times as you desire. Don’t be buffaloed by the buffaloes. I heard a herd ahead.
88
LIFE SENTENCES
[1] Explain the meaning of the following sentence.
Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
[2] Punctuate the following sentence so it makes sense.
John where James had had had had had had had had had had had the teacher’s approval.
[3] On what day was the following sentence written?
When the day after tomorrow is yesterday then ‘today’ will be as far from Sunday as that day was which was ‘today’ when the day before yesterday was tomorrow.
The next problem concerns a much simpler sentence.
89
IN THE BEGINNING (AND THE MIDDLE AND THE END) WAS THE WORD
Which single word can be placed in each of the eight marked spaces to make eight meaningful sentences?
_ I kicked him in the leg yesterday.
I _ kicked him in the leg yesterday.
I kicked _ him in the leg yesterday.
I kicked him _ in the leg yesterday.
I kicked him in _ the leg yesterday.
I kicked him in the _ leg yesterday.
I kicked him in the leg _ yesterday.
I kicked him in the leg yesterday _.
The following puzzles concern the shapes of letters, words and sentences. Just my type of problem.
90
LOOKING AT LETTERS
[1] The single ‘l’ and two ‘z’s shown below make up a picture of which letter?
[2] What symbol comes next?
[3] Decipher these three common English words.
91
A MATTER OF REFLECTION
WHEN THE AUTHOR WROTE THIS QUESTION, THEY HID A WORD IN IT THAT HAS A SPECIAL PROPERTY. IT IS THE ONLY WORD IN THE TEXT THAT LOOKS EXACTLY THE SAME WHEN YOU READ IT IN A MIRROR WITH THE PAGE TURNED UPSIDE DOWN. ALL THE OTHER WORDS WILL HAVE AT LEAST ONE LETTER THE WRONG WAY ROUND. CAN YOU DISCOVER THE SPECIAL WORD WITHOUT USING A MIRROR?
92
THE BLANK COLUMN
When the title of this book, shown below, is typed repeatedly in the same paragraph, a peculiar phenomenon is visible. A blank column (marked in grey) runs through the text.
Would a blank column appear when any sentence is repeated ad infinitum, or did I deliberately choose the title because it creates this strange effect? (Consider only sentences that are shorter than the length of the line.)
By the time you opened this book, you had already seen a puzzle based on the outline of a letter. The shape emblazoned in white and blue on the front cover is a capital letter cut out of paper and folded. The challenge is to deduce the original letter, which you are told is not the first one that comes to mind. This puzzle has pedigree. It was the first original problem devised by Scott Kim, then aged 12, wh
o would go on to become one of the most important puzzle designers in the US. One of Kim’s trademarks is to take an idea and explore its many variations.
93
WELCOME TO THE FOLD
The strange-looking alphabet shown opposite has been created by cutting 26 standard-looking capital letters from a piece of paper and folding each of them once. For example, the A is actually an H.
Can you unfold the other 25 letters to discover which letter of the alphabet each one actually is?
Kim’s interest in letters and typefaces flourished while he was an undergraduate at Stanford University in the mid-1970s, where he studied mathematics, music and computer science. Fascinated by the intersection of art and technology, and particularly user-interface design, he found an innovative way to link his mathematical and artistic sides: by devising a symmetrical calligraphy in which words can be read in more than one way. His renderings of the words below, for example, read the same if you turn them upside down.
Words which can be read in different ways are called ‘ambigrams’. (Kim is considered a co-inventor of the ambigram, together with the artist John Langdon, who started doing similar work independently at around the same time.) Kim has designed hundreds of ingenious ambigrams, and is such a master that if you give him any word he can pretty much turn it into an ambigram on the spot.
The most common type of ambigram (like the examples shown above) uses 180-degree ‘rotational symmetry’ – that is, when you rotate it half a turn the word is the same. The capital letters I, N, O and S have rotational symmetry, so, depending on the typeface, a word like is a natural ambigram. As is and, in lower case,
Creating an ambigram is an entertaining challenge. It requires you to think about how much visual information you need to suggest a letter of the alphabet. You might need to remove visual elements from some letters, but not so much that the letter is unrecognisable, or you might need to add visual elements, but not so much that the eye is overwhelmed. The form relies on trickery and suggestion. You can contaminate a letter with quite a lot of extraneous detail the eye will choose not to see if the salient features are there.
The next puzzle is Scott Kim’s introduction to how to draw in this style.
94
MY FIRST AMBIGRAM
Find a way to write the following words so that they read the same upside down:
USA
VISTA
CHILD
These three words present a perfect initiation into the pain and pleasure of ambigram design. They already have some helpful symmetries. Try to come up with several solutions for each word. There is no ‘correct’ answer, but the more you do it, the more elegant your solution will become. You might have to stick with capitals, or use lower-case letters, or use a mixture of the two.
Your next ambigram challenge is to write your name. Lucky for you, Bob, Una and Wim. Sorry Aloysius, Josephine and Waldemar.
Scott Kim is most famous for his ambigrams, but he has also designed a huge number and variety of excellent puzzles. Some of my favourites are based on the shape and form of letters. In the next one, he asks us to recreate words with almost no information at all.
95
BOXED PROVERBS
Can you decipher these 10 familiar sayings? Each letter in the saying has been replaced by a black box the same height and width of that letter.
This puzzle is also a kind of optical illusion. If you squint at the page, the phrases become easier to see. Our eyes glide first to the letters that stick up or down. Even though we read from left to right, reading is made easier thanks to the up and down rhythm of ascenders and descenders, which is why it is easier to read large amounts of text in lower case than in upper case.
Scott Kim’s boxed proverbs puzzle is a variation of a similar pattern-recognition problem in which we are given only the first letter of the word.
96
NMRCL ABBRVTNS
Identify the words in each of the abbreviated statements below. Each word is denoted by its initial letter (in upper case), and the statements are all true. For example, the answer to ‘A H. has 5 F.’ would be ‘A hand has 5 fingers’.
[1] W. + S. + S. + A. = the 4 S.
[2] A. + A. + A. + A. + E. + N. A. + S. A. = the 7 C.
[3] S. + G. + D. + B. + S. + D. + H. = the 7 D.
[4] The A. has 26 L.
[5] A G. has 6 S.
[6] There are 52 C. in a D.
[7] A S. has 8 L.
[8] A C. has 64 S.
[9] A G. C. has 18 H.
[10] A 1 followed by 6 Z. is a M.
The following question about a Russian family also concerns initials.
To answer it, however, you need to know about the Russian system of patronymics, in which a man’s middle name is his father’s name with the suffix -ovich, -evich or –ich added. For example, in Russia I would be Alex Davidovich Bellos, since my father’s name is David, and my sons would both have the middle name Alexovich.
97
THE NAME OF THE FATHER
Here are names of males in a Russian family. The first initial is the first name, and the second initial is the patronymic. In this family every father has two sons, the patriarch of the family has four grandsons, and his sons have two grandsons each.
A. N. Petrov K. T. Petrov N. K. Petrov
B. M. Petrov M. M. Petrov N. T. Petrov
G. K. Petrov M. N. Petrov T. M. Petrov
K. M. Petrov N. M. Petrov
Draw the Petrov family tree.
From Russia, let’s take a day trip across the border to Estonia. The capital city is Tallinn, and the national language is Estonian, which is spoken by 1.1 million people and closely related to Finnish.
If you’re an Estonian speaker, the next puzzle will be very easy. If you’re not, you will learn something curious about how Estonians tell the time.
98
TELLING THE TIME IN TALLINN
The times on the following clock faces are written in Estonian.
If an Estonian looks at his watch and tells you one of the following, what time is it?
[1] Kakskümmend viis minutit üheksa läbi.
[2] Kolmveerand kaksteist.
[3] Pool kolm.
[4] Veerand neli.
[5] Kolmkümmend viis minutit kuus läbi.
What’s the Estonian for these times?
[1] 4.15 [2] 8.45 [3] 11.30 [4] 7.05 [5] 12.30
You will need to know the following Estonian number words:
6 kuus 7 seitse 8 kaheksa 10 kümme
The Tallinn time teaser is adapted from a question that appeared in the North American Computational Linguistics Olympiad, a wonderful competition in which secondary-age students solve linguistics puzzles.
Linguistics puzzles require a mixture of deductive logic, code-breaking skills and a sense of how languages work. When you tackle them you feel like a detective deciphering an ancient scroll. Often you have to stab in the dark. Whereas a purely mathematical puzzle might introduce you to a new concept, a linguistics puzzle will often reveal fascinating peculiarities of different languages or cultures.
Here’s another puzzle adapted from the North American Computational Linguistics Olympiad. It concerns the Waorani, a group of about 4,000 Amerindians who live in the Ecuadorian Amazon. The Waorani language is an isolate – that is, it is not known to be related to any other language.
99
COUNTING IN THE RAINFOREST
The following equations involve the numbers from 1 to 10 in the Waorani language. Each underlined sequence represents a different number, and the symbols +, × and 2 are used in their normal senses of addition, multiplication and squaring.
[1] mẽña mẽña mẽña mẽña + mẽña go mẽña = ãẽmãẽmpoke go aroke × 2
[2] aroke2 + mẽña2 = ãẽmãẽmpoke
[3] ãẽmãẽmpoke go aroke2 = mẽña go mẽña × ãẽmãẽmpoke mẽña go mẽña
[4] mẽña × ãẽmãẽmpoke = tipãẽmpoke
What are t
he numbers from 1 to 10 in Waorani?
You don’t need to cross the world to find an impenetrable language. Just find an organic chemist.
100
CHEMISTRY LESSON
Match the chemical formulae to their names.
C3H8, C4H6, C3H4, C4H8, C7H14, C2H2
Heptene, Butyne, Propane, Butene, Ethyne, Propyne.
Well, that was a gas!