Hints - The GCHQ Puzzle Book (2016)

The GCHQ Puzzle Book (2016)

Penguin Books


Many of the puzzles in this book may look more intractable than they really are. In a lot of cases the trick is to approach the puzzle with the right mindset - in other words to think like a GCHQ puzzle setter. So for example when looking at a set of words, don’t think of them as words, think of them as a set of letters. And when looking at a set of letters, think whether they could be converted into numbers. And when looking at a set of numbers, think whether they could be converted into letters.

Here are some examples:

What do the members of each set below have in common:



What is the final member of these sequences?

3. D, H, L, P, T, ?

4. 1, 14, 19, 23, 5, ?

In question 1 there isn’t really anything Bolivia and a piccolo have in common. But if you look at the words as a set of letters you will see they all have seven letters. That is something they have in common, but it’s not particularly unusual. But if you look more closely you will see that each word ends in a vowel. That’s a bit more unusual, and is on the right lines. Having spotted this about the last letters, have a look at the first letters - and you will notice that in each case the first letter of the word comes immediately after the last letter of the word in the alphabet. That is definitely unusual, and now you have the complete answer.

In question 2, again there isn’t really anything Illinois and pitiless have in common. So look closely at the words - they all have 8 letters. But what else have they in common? There is nothing particularly special about the first letters E, I, P, S - or the last letters E, S, S, E. How about every other letter - ETAE, ILNI, PTLS, SRGL? Again nothing particularly special there. So try looking inside the words. Do they contain hidden words? Well, ESTIMATE contains TIM and MAT, and ILLINOIS contains ILL, LIN and LINO. PITILESS contains several shorter words, but STRUGGLE only really contains RUG. Now you see the connection: the words contain MAT, LINO, TILES and RUG - all things you can put on a floor.

For questions 3 and 4, the letters in 3 don’t make a word, and the numbers in 4 don’t seem to be a numerical sequence. One thing to try is to convert letters to numbers and vice versa based on where they are in the alphabet. By this method A=1, B=2, C=3, D=4, E=5, F=6, G=7, H=8, I=9, J=10, K=11, L=12, M=13, N=14, O=15, P=16, Q=17, R=18, S=19, T=20, U=21, V=22, W=23, X=24, Y=25 and Z=26.

Using this conversion on the questions turns them into:

3. 4, 8, 12, 16, 20, ?


4. A, N, S, W, E, ?

Now the answers are clear. Question 3 is every 4th letter in the alphabet, so X is the final member. And question 4 spells out ANSWER, so the final member is 18, representing the R.

This sort of approach will help with many of the questions. Look at first letters, last letters, central letters, every other letter. See if you can add a letter to the front, or at the end, or in the middle, or if you can remove a letter somewhere. Look for hidden words, or hidden words in reverse. Look for the letters of the alphabet running through a sequence.

When converting letters to numbers and vice versa, there are ways other than A=1, B=2, etc. Letters could be converted into their Scrabble score (see Appendix), or possibly via their positions on a typewriter keyboard or mobile phone keypad (ABC=2, DEF=3, GHI=4, JKL=5, MNO=6, PQRS=7, TUV=8, WXYZ=9).

Letters may need to be interpreted as their NATO phonetic alphabet equivalents: A=ALFA, B=BRAVO, etc (see Appendix).

Some number puzzles in the book may be mathematical, but they may also be based on how the numbers are written as words (ONE, TWO, THREE, etc). The British method of writing numbers is used in this book so 108 would be written ONE HUNDRED AND EIGHT. Numbers may also refer to elements in the periodic table, or their symbols (see Appendix).

As well as English we use some French and German, particularly the numbers:



And we also use Roman numerals: I, II, III, IV, V, VI, VII, VIII, IX, X, …

(I=1, V=5, X=10, L=50, C=100, D=500, M=1000)

There are a number of common themes which run through this book, and having these in mind will help with many of the questions. Some of these - The Periodic Table, US states and US presidents, NATO phonetic alphabet, the Braille alphabet, the Greek alphabet, Morse code and Scrabble values - are listed in the appendix. We are also fond of some shorter sets:

Cardinal Points: North, East, South, West

Rainbow: Red, Orange, Yellow, Green, Blue, Indigo, Violet

Tonic Sol-fa scale: Do, Re, Mi, Fa, So, La, Ti

The 12 days of Christmas: 12 Drummers drumming, 11 Pipers piping, 10 Lords a-leaping, 9 Ladies dancing, 8 Maids a-milking, 7 Swans a-swimming, 6 Geese a-laying, 5 Gold rings, 4 Calling birds, 3 French hens, 2 Turtle doves and a Partridge in a pear tree

Nine Muses: Calliope, Clio, Erato, Euterpe, Melpomene, Polyhymnia, Terpsichore, Thalia, Urania

Snow White’s Seven Dwarfs: Bashful, Doc, Dopey, Grumpy, Happy, Sleepy, Sneezy

Planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune (and, until recently, Pluto)

Zodiac: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Pisces

Chinese zodiac: Rat, Ox, Tiger, Rabbit, Dragon, Snake, Horse, Goat, Monkey, Rooster, Dog, Pig

Mohs scale of hardness: Talc (1), Gypsum (2), Calcite (3), Fluorite (4), Apatite (5), Orthoclase feldspar (6), Quartz (7), Topaz (8), Corundum (9), Diamond (10)

There are also larger sets we like: Olympic host cities, London tube lines, countries and capital cities, books of the Bible, symphonies and their composers; days of the week, months of the year, and animals may make an appearance too.

You may be able to tell something about our literary and musical tastes from the questions. We read Shakespeare, Tolkien, A A Milne, Enid Blyton’s Famous Five books and JK Rowling’s Harry Potter books. We watch James Bond films, Dr Who and Monty Python. We listen to the Beatles and the Rolling Stones.

In our leisure time we play Scrabble, Monopoly and Cluedo, but also Darts and Snooker. So as well as Scrabble scores for letters you may come across Monopoly properties, Cluedo characters, weapons and rooms; numbers around a Dartboard; and the values of Snooker balls.

There are a few phrases we like, and one particular word:

To be or not to be, that is the question

If music be the food of love, play on

The quick brown fox jumps over the lazy dog


Word origins (i.e. which language they come from) and rhyming words come up too.

We also like writing words as tables and reading other words from them. So for example:

5.    What series do the following words convey: DORIC, IDEAL, TIMID, ALOFT, ESSAY?

As nothing stands out try putting the words in a table:


If you then read letters horizontally and vertically, starting top left, you can find DO, RE, MI, FA, SO, LA, TI - with TI positioned so that DO can be read off again next:


There are also word squares where the same words can be read across and down.

As you would expect from a GCHQ puzzle book there are questions which involve breaking codes. By far the most common is simple substitution cipher. In this method each letter in a message is replaced by another particular letter. You have to work out which letter is replaced by which, and this could be done by doing a frequency count of the letters in a message (you would expect the commonest letters to be those representing E, T, A, O, I, N …) or by looking for common words (A, THE, AND, etc). Most substitution ciphers rely on a keyword or keyphrase which can be discovered once you know what the substitution alphabet is. This is most easily explained with an example:


The plain text has been enciphered using this substitution alphabet:


The keyword here is SEMAPHORE. This has been written first (apart from the second E) followed by the rest of the alphabet in order. This method can mean that letters towards the end of the alphabet are enciphered to themselves (TUVWXYZ in this case), which can be a clue to solving such ciphers.

As you would also expect from a GCHQ puzzle book there are some mathematical puzzles - but only recreational maths knowledge should be needed. Some series used are:

Triangular numbers: 1, 3, 6, 10, 15, 21, etc

Square numbers: 1, 4, 9, 16, 25, 36, etc

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc

Powers of two: 1, 2, 4, 8, 16, 32, 64, 128, etc

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, etc (each number is the sum of the previous 2)

Pi (3.14159265358979 …) comes up a few times, and e (2.71828182845 …) makes an appearance too.

Sometimes different bases are used. Popular ones are:

Binary (base 2): 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, etc

Hex (base 16): 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, etc

Other bases are sometimes used too.

Finally, as well as the Where? and Which? questions which are explained here, there are some other regular questions:

A to Z: In these questions the letters A to Z before the ‘=’ represent single-word members of a particular set of items which start with these letters, and the letters after the ‘=’ represent the initials of people or things which form all or part of the member of the set.

Chains: In these questions a chain can be built up from pairs of words or names which overlap in some way. You have the first and last members of the chain and have to work out the missing links.

Sums: In Sums questions you are presented with a strange-looking sum. Sometimes it consists of words, and sometime pictures. For these you need to find a number associated with each word or picture - and these associations will be of the same kind in each question. You need to identify the numbers and then calculate the answer to the sum. The answer will also have a number associated with it in the same way, and the answer to thequestion will be a word or picture of the same kind as those in the question.

For example:

(evens × ether - eon) / tow = ?

To solve this you need to spot the connection between all the words - which is that they are all anagrams of numbers. So using the associated numbers gives:

(7 × 3 - 1) / 2 = 10

Ten is an anagram is anagram of net - so the answer to the question is ‘net’.