by Alex Bellos
We turn over our first slip. Now randomly generate a number, N.
Our strategy is now as follows: if N is bigger than the number on the first slip, choose the second slip, but if N is lower than the number on the first slip, choose the first slip.
There are three possibilities for N. It could be bigger than both A and B, in which case there is a 1 in 2 chance of choosing the correct slip, as shown below.
N > A > B
Slip 1 Slip 2 Choice
A B N > A, so B
B A N > B, so A
Or N is smaller than both A and B, in which case there is still a 1 in 2 chance of choosing the correct slip:
A> B > N
Slip 1 Slip 2 Choice
A B N < A, so A
B A N < B, so B
However, if N is between A and B, then the strategy gets us the right slip every time:
A > N > B
Slip 1 Slip 2 Choice
A B N < A, so A
B A N > B, so A
If there is a non-zero chance that our randomly generated number N is between A and B, our chances of picking the right number are better than 1 in 2. And there must be a non-zero chance, since whatever N is there will always be numbers either side of it, and these could be A and B.
The solution to the problem when there are more than three slips of paper is to turn over the first 37 per cent of the slips, and then choose the first slip with a number higher than any you have seen before. The percentage is the number 1⁄e, where e is the exponential constant, which is 2.718 to three decimal places. Unfortunately, the proof is too detailed to include here.
123 THE THREE PRISONERS
Neither of the prisoners reasoned correctly. The odds of A getting the pardon remain 1 in 3, but the odds of C getting the pardon increase to 2 in 3.
At the beginning of the problem, the governor picks one of the prisoners at random to be pardoned, which means that each prisoner has a 1 in 3 chance of being pardoned.
When the governor tells A that B will be executed, A’s chances of being pardoned remain at 1 in 3, because whichever prisoner the governor decided to pardon, she will always be able to give A the name of another prisoner who will be executed. The fact that she said B in this case provides A with no useful information about his fate.
However, if the chances of A being pardoned remain at 1 in 3, the chances of A not being pardoned must be 2 in 3. In other words, the chance of B or C being pardoned is 2 in 3. However, we know that B will be executed. So the chance of C being pardoned must be 2 in 3.
124 THE MONTY FALL PROBLEM
It’s not in your advantage to switch. The chance of the car being behind door No. 1 and the chance of it being behind door No. 3 is 1 in 2 in both cases.
Let’s look at this by drawing up a table of equally likely outcomes, and seeing what happens when you stick or switch. You have chosen door No. 1. There are six equally likely cases to consider, since there are two possible ways that Monty can fall for each possible location of the car.
Door concealing car Door Monty opens Stick Switch
1 2 Win Lose
1 3 Win Lose
2 2 * *
2 3 Lose Win
3 2 Lose Win
3 3 * *
When accounting for all possible outcomes, we must include the times when Monty falls and opens the door concealing the car. I’ve marked these in the table with a *. We can ignore these rows in our calculations, however, since we know that the door opened to reveal a goat.
We can see that if Monty falls and opens a door concealing a goat, we will win by sticking in 2 cases out of 4, and win by switching in 2 cases out of 4. In other words, there is no advantage in switching.
The process by which Monty opens the door is crucial. In the original Monty Hall problem, he knows where the car is, and his intention is to open a door that reveals a goat. This set-up means that it’s in your advantage to switch. In the Monty Fall problem, however, Monty arbitrarily opens a door that just happens to conceal a goat. To work out the correct probabilities, we need to take account of all the times he could have arbitrarily opened a door to reveal the car.
125 RUSSIAN ROULETTE
In adjacent chambers, it’s best to stick. In non-adjacent chambers, it’s best to spin.
There are six chambers. Let’s call them 1, 2, 3, 4, 5 and 6.
The adjacent case. The bullets are, say, in 1 and 2. When the cylinder is spun there is a 4⁄6 chance it will stop with a bullet-free chamber lined up with the barrel. If your captor spins again, the chances of getting a bullet-free chamber remain 4⁄6, or 66 per cent. If you stick, however, and the cylinder turns to the next chamber, positions 3, 4, 5 and 6 become positions 4, 5, 6, and 1. Three of those four are bullet-free chambers, giving a chance of survival of 75 per cent. You should stick.
The non-adjacent case. The bullets are, say, in 1 and 4. If you stick now, the bullet-free positions 2, 3, 5 and 6 become 3, 4, 6 and 1. Only two of those four are bullet-free chambers, giving you only a 50 per cent chance of survival. Since spinning gives you a 66 per cent chance of survival, you should spin.
A list of the puzzles and a note on their sources
The list below contains the sources – books, magazines, websites and friends – for the puzzles in this book. Some of the texts listed are not the original sources; it’s often hard to find out exactly where a puzzle first emerged. If an original source is known, it can usually be found in David Singmaster’s extensive and thorough Sources in Recreational Mathematics, available online, which I consulted on almost a daily basis. Every attempt has been made to contact copyright holders. All queries should be addressed to the publisher.
Tasty teasers
Number conundrums
1 Ian Stewart, Professor Stewart’s Hoard of Mathematical Treasures, Profile Books, 2009.
2 Martin Gardner, The Unexpected Hanging and Other Mathematical Diversions, University of Chicago Press, 1969.
3 Boris A. Kordemsky, The Moscow Puzzles, Dover Publications, 1992.
4, 5, 6 Nobuyuki Yoshigahara, Puzzles 101, A. K. Peters/CRC Press, 2004.
The puzzle zoo
ANIMAL PROBLEMS
1 The Three Rabbits. Traditional
2 Dead or Alive. The Family Friend, 1849.
3 Good Neighbours. Des MacHale and Paul Sloane, Hall of Fame Lateral Thinking Puzzles, Sterling, 2011.
4 A Fertile Family. Based on an idea from www.bio.miami.edu/hare/scary.html.
5 A Bunch of Hops. Ron Knott, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpuzzles.html
6 Crossing the Desert. Adapted from Pierre Berloquin, The Garden of the Sphinx, Barnes & Noble, 1996.
7 Save the Antelope. Adapted from Pierre Berloquin, The Garden of the Sphinx, Barnes & Noble, 1996.
8 The Thirteen Camels. David Singmaster, Sources in Recreational Mathematics, South Bank University, 1991.
9 Camel vs Horse. Traditional.
10 The Zig-zagging Fly. Traditional.
11 The Ants on a Stick. Told to me by Rob Eastaway. www.robeastaway.com
12 The Snail on the Elastic Band. Martin Gardner, Time Travel and Other Mathematical Bewilderments, W. H. Freeman, 1988.
13 Animals that Turn Heads. Kobon Fujimura, The Tokyo Puzzles, Frederick Muller, 1979. Nobuyuki Yoshigahara, Puzzles 101, A. K. Peters/CRC Press, 2004.
14 Banishing Bugs From the Bed. Peter Winkler, Mathematical Mind-Benders, A. K. Peters/CRC Press, 2007.
15 The Dumb Parrot. Yuri B Chernyak and Robert M Rose, The Chicken from Minsk, Basic Books, 1995.
16 Chameleon Carousel. Question first posed in the International Tournament of the Towns, 1984.
17 The Spider and the Fly. Henry Ernest Dudeney, 536 Curious Problems & Puzzles, Barnes & Noble, 1995.
18 The Meerkat in the Mirror. First told to me by Carlos Vinuesa.
19 Catch the Cat. First told to me by Charlie Gilderdale.
20 Man Spites Dog. Des MacHale and Paul Sloane, Hall of Fame Lateral Thinking Puzzle
s, Sterling, 2011.
21 The Germ Jar. Naoki Inaba, ‘Numberplay’ column, New York Times.
22 The Fox and the Duck. Martin Gardner, Mathematical Carnival, The Mathematical Association of America, 1989.
23 The Logical Lions. Derrick Niederman, Math Puzzles for the Clever Mind, Puzzlewright Press, 2013.
24 Two Pigs in a Box. Steven E. Landsburg, Can You Outsmart an Economist, Mariner Books, 2018
25 Ten Rats and One Thousand Bottles. First heard on the YouTube channel PBS Infinite Series.
Tasty teasers
Gruelling grids
1, 2 Carlos D’Andrea, University of Barcelona.
3 www.wesolveproblems.org.uk
4, 5, 6 Daniel Finkel, www.mathforlove.com
I’m a mathematician, get me out of here
SURVIVAL PROBLEMS
26 Fire Island. Richard Wiseman’s Friday Puzzle. https://richardwiseman.wordpress.com/2012/07/02/6488/
27 The Broken Steering Wheel. Adapted from Rob Eastaway and David Wells, 100 Maddening, Mindbending Puzzles, Portico, 2018
28 Walk the Plank. Adapted from Henry Dudeney, 536 Curious Problems & Puzzles, Barnes & Noble, 1995.
29 The Three Boxes. Adapted from Raymond Smullyan, What is the Name of this Book?, Dover Publications, 1978.
30 Safe Passage. Simon Singh, The Code Book, Fourth Estate, 1999.
31 Crack the Code. https://puzzling.stackexchange.com/questions/46871/crack-thelock-code.
32 Guess the Password. ‘Technical Problems’, from MIT’s The Tech, April 17, 2005.
33 The Spinning Switches. Peter Winkler, Mathematical Puzzles, A Connoisseur’s Collection, A. K. Peters/CRC Press, 2004.
34 Protect the Safe. Pierre Berloquin, The Garden of the Sphinx, Barnes & Noble, 1996.
35 The Secret Number. Steven E. Landsburg, Can You Outsmart an Economist?, Mariner Books, 2018
36 Removing the Handcuffs. Marin Gardner, Mathematics, Magic and Mystery, Dover Publications, 1956.
37 The Reversible Trousers. Martin Gardner, Sixth Book of Mathematical Diversions from Scientific American, University of Chicago Press, 1984.
38 Mega Area Maze, By Naoki Inaba.
39 Arrow Maze. Mathematical Olympiads 1999–2000: Problems and Solutions from Around the World, Mathematical Association of America, 2002
40 The Twenty-Four Guards. Adapted from Raymond Smullyan, What is the Name of this Book?, Dover Publications, 1978.
41 The Two Envelopes. Futility Closet, 2009, https://www.futilitycloset.com/2009/08/05/royal-pain/
42 The Missing Number. Peter Winkler, Mathematical Puzzles, A Connoisseur’s Collection, A. K. Peters/CRC Press, 2004.
43 The One Hundred Challenge. Rob Eastaway and David Wells, 100 Maddening, Mindbending Puzzles, Portico, 2018
44 The Fork in the Road. Martin Gardner, My Best Mathematical and Logic Puzzles, Dover Publications, 1994.
45 Bish and Bosh. The Fork in the Road. Martin Gardner, My Best Mathematical and Logic Puzzles, Dover Publications, 1994.
46 The Last Request. Raymond Smullyan, The Riddle of Scheherazade, A. A. Knopf, 1997
47 The Red and Blue Hats. ‘Mathematics in Education and Industry’, Maths Item of the Month, August 2010. https://mei.org.uk/month_item_10#aug
48 The Majority Report. Peter Winkler, Mathematical Puzzles, A Connoisseur’s Collection, A. K. Peters/CRC Press, 2004.
49 The Room with the Lamp. Peter Winkler, Mathematical Puzzles, A Connoisseur’s Collection, A. K. Peters/CRC Press, 2004.
50 The One Hundred Drawers. Peter Winkler, Mathematical Mind-Benders, A. K. Peters/CRC Press, 2007.
Tasty teasers
Riotous riddles
1 Traditional.
2 Raymond Smullyan.
3, 4, 5, 6, 7, 8, 9, 10 Adapted from Hall of Fame Lateral Thinking Puzzles, by Sloane and MacHale, Sterling, 2011.
Cakes, cubes and a cobbler’s knife
GEOMETRY PROBLEMS
51 The Box of Calissons. Guy David and Carlos Tomei, ‘The Problem of the Calissons’, The American Mathematical Monthly, vol. 96, 1989.
52 The Nibbled Cake. Source unknown.
53 Cake for Five. Source unknown.
54 Share the Doughnut. https://www.mathsisfun.com/puzzles/horace-and-thedoughnut.html
55 A Star is Born. Edward B. Burger, Making Up Your Own Mind, Princeton University Press, 2018
56 Squaring the Rectangle. A version appears in the Wakoku Chie-Kurabe, 1727, and repeated by many other authors since then.
57 The Sedan Chair. Mathematical Puzzles of Sam Loyd, Dover, 1959.
58 From Spade to Heart. Mathematical Puzzles of Sam Loyd, Dover, 1959.
59 The Broken Vase. Pierre Berloquin, The Garden of the Sphinx, Barnes & Noble, 1996.
60 Squaring the Square. Derrick Niederman, Math Puzzles for the Clever Mind, Puzzlewright Press, 2013.
61 Mrs Perkins’s Quilt. Henry Ernest Dudeney, Amusements in Mathematics, 1917.
62 The Sphinx and Other Reptiles. Author’s own.
63 Alain’s Amazing Animals. http://en.tessellations-nicolas.com
64 The Overlapping Squares. Pierre Berloquin, The Garden of the Sphinx, Barnes & Noble, 1996.
65 The Cut-Up Triangle. Nobuyuki Yoshigahara, Puzzles 101, A. K. Peters/CRC Press, 2004.
66 Catriona’s Arbelos. Catriona Shearer. https://twitter.com/cshearer41
67 Catriona’s Cross. Catriona Shearer. https://twitter.com/cshearer41
68 Cube Angle. Kobon Fujimura, The Tokyo Puzzles, Frederick Muller, 1979.
69 The Menger Slice. As told to me by George Hart. https://www.georgehart.com
70 The Peculiar Peg. Martin Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, University of Chicago Press, 1961.
71 The Two Pyramids. Peter Winkler, Mathematical Puzzles, A Connoisseur’s Collection, A K Peters, 2004.
72 The Rod and the String. Trends in International Mathematics and Science Study, 1995.
73 What Colour is the Beard? Author’s own.
74 Around the World in 18 Days. Jules Verne, Around the World in Eighty Days, 1873.
75 A Whisky Problem. http://mathforum.org/wagon/2014/p1191.html
Tasty teasers
Pencils and utensils
1 Fredrik Cattani, via email. http://www.filiokusfredrik.se
2 Pierre Berloquin, 100 Games of Logic, Barnes & Noble, 1977.
3 ‘Good Hands’, Futility Closet, 2012. https://www.futilitycloset.com/2012/04/30/good-hands/
4 Fredrik Cattani, via email, http://www.filiokusfredrik.se
5 Source unknown.
A wry plod
WORD PROBLEMS
76 The Sacred Vowels. Adapted from Christian Bok, Eunoia, Canongate, 2009.
77 Winter Reigns. Word Ways, February, 1974.
78 Five Deft Sentences. 1., 3., 5., Author’s own. 2. https://www.grammarly.com/blog/16-surprisingly-funny-palindromes/ 4. Dmitri Borgmann, Language on Vacation, Charles Scribner’s Sons, 1965.
79 The Consonant Gardener. Author’s own and Martin Gardner, Sixth Book of Mathematical Diversions from Scientific American, University of Chicago Press, 1984.
80 Kangaroo Words. Chris Smith’s Maths Newsletter, issue 476.
81 The Ten-Letter Words. Author’s own.
82 Ten Notable Numbers. Sources unknown.
83 The Questions That Count Themselves. Lee Sallows. http://www.leesallows.com
84 The Sequence That Describes Itself. Eric Angelini.
85 Sexy Lexy. Based on an idea from Eric Angelini.
86 Letters in a Box. Martin Gardner, Sixth Book of Mathematical Diversions from Scientific American, University of Chicago Press, 1984; Peter Winkler, Mathematical Mind-Benders, A. K. Peters/CRC Press, 2007.
87 Wonderful Words. Various sources, including Martin Gardner, Mind-Boggling Word Puzzles, Sterling, 2001; and Henry Ernest Dudeney, The World’s Best Word Puzzles, 1925.
88 Life Sentences. 1.,2. Widely known. 3. More Mathematical Puzzles of Sam Loyd,
Dover Publications, 1960.
89 In the Beginning (and the Middle and the End) was the Word. Martin Gardner, Mind-Boggling Word Puzzles, Sterling, 2001.
90 Looking at Letters. Peter Winkler, Mathematical Mind-Benders, A. K. Peters/CRC Press, 2007, and Martin Gardner, Mind-Boggling Word Puzzles, Sterling, 2001.
91 A Matter of Reflection. Adapted from Rob Eastaway and David Wells, 100 Maddening, Mindbending Puzzles, Portico, 2018
92 The Blank Column. Martin Gardner, Wheels, Life and Other Mathematical Amusements, W. H. Freeman, 1983.
93 Welcome to the Fold. Scott Kim.
94 My First Ambigram. Scott Kim.
95 Boxed Proverbs. Scott Kim.
96 Nmrcl Abbrvtns. Edwin F. Meyer and Joseph R. Luchsinger, Book of Puzzles, Gedanken Publishing, 2012.
97 The Name of the Father. 200 Problems in Linguistics and Mathematics, 1972, as translated in Tanya Khovanova’s Math Blog https://blog.tanyakhovanova.com.
98 Telling the Time in Tallinn. Adapted from Babette Newsome, North American Computational Linguistics Olympiad, 2014, online practice problem, www.nacloweb.org.
99 Counting in the Rainforest. Adapted from Dragomir Radev, North American Computational Linguistics Olympiad, 2012, online practice problem, www.nacloweb.org.
100 Chemistry Lesson. www.lingling.ru, as translated in Tanya Khovanova’s Math Blog https://blog.tanyakhovanova.com
Tasty teasers
Bongard bafflers
Sample: Bongard Problem 6 by Mikhail Bongard.
1 Bongard Problem 40 by Mikhail Bongard.
2 Bongard Problem 44 by Mikhail Bongard.
3 Bongard Problem 29 by Mikhail Bongard.
4 Bongard Problem 180 by Harry Foundalis.
Sleepless nights and sibling rivalries
PROBABILITY PROBLEMS
