Farewells are always poignant: think of Romeo taking his leave of the fair Juliet, tragically oblivious to narrative imperative; Doctor Watson reporting “with a heavy heart” the (presumed) demise of his friend at the Reichenbach Falls; Winnie the Pooh’s emotional goodbyes to Christopher Robin—
I’ll get me coat. Indeed, I will, for this is my last “Enigmas & puzzles” column.
Enigmas made its debut in January 2003. Before then I’d been writing the “Mathematical recreations” column for Scientific American, following in the footsteps of the inimitable Martin Gardner, a journalist with strong interests in stage magic and puzzles. In 1956 he wrote an article for the magazine about a new mathematical game, flexagons, which was such a success with readers that the editor asked him to write a regular column on mathematical games. It ran until 1981, and was written in an engaging and comprehensible style that won him thousands of fans, myself among them.
Puzzles don’t appeal to everyone, but there’s a loose-knit community of enthusiasts—some of them mathematicians—who relish the intellectual challenge of a new problem, trade ideas, find ingenious ways to “cook” a puzzle (find a solution that has been overlooked, or a loophole in the wording), and post puzzles on Twitter. Among them are John Horton Conway, who invented the “game of life” in which black and white dots obeying simple rules behave in amazingly complex ways, and Persi Diaconis and Ron Graham, who recently wrote a book on the mathematics of magic tricks. All three are distinguished research mathematicians.
Even so, it’s tempting to see puzzles as a rather trivial pastime. The great British mathematician Godfrey Harold Hardy surely thought so. In his 1940 essay “A Mathematican’s Apology,” he remarks that only four numbers (153, 370, 371, and 407) equal the sum of the cubes of their digits, writing: “These are odd facts, very suitable for puzzle columns... but there is nothing in them which appeals much to a mathematician.” It’s true that such things lack deep significance, but bare-hands messing around with simple concepts in an interesting context is an excellent way to develop basic mathematical abilities.
It worked for me. As a kid, my friends and I invented elaborate games with complicated rules—tiddlybowls, a tabletop version of bowls with tiddlywinks; Red Fred (nothing like blackjack); a massive extension of the Buccaneer board game which occupied an entire room. We once tossed five dice 100,000 times to spot statistical patterns. It took weeks. When I was 14 or thereabouts I discovered Martin Gardner, and Scientific American became a fixture in our house. Persi Diaconis has remarked (not in quite these words) that Gardner turned hundreds of children into mathematicians, and hundreds of mathematicians into children. He did both to me.
What I most like about puzzles is the challenge of finding the answer. Mathematical research at the frontiers is really just an elaborate exercise in solving puzzles. The ingredients—partial differential equations, seven-dimensional spheres, and so on—are more esoteric, but the thought processes are surprisingly similar, in general terms. When you get stuck on a research problem it’s just like getting stuck on a puzzle, and would-be puzzle solvers can use the same techniques to unstick themselves. I try to find simpler versions of the same problem, hoping to see some general principle. I look at specific examples. When I’m really stuck, I go off and do something else. Often inspiration then strikes.
There was a course I used to teach to undergraduate students—“Applied sources of pure mathematics”—that provided opportunities to set serious technical problems in puzzle format, such as designing a two-way light switch, which is really an exercise in mathematical logic. In fact, behind many an Enigma lurks a significant mathematical idea. Puzzles can present a difficult and daunting subject in a friendly and amusing way, a realisation that goes back at least 3,500 years to ancient Babylon where schoolboys were taught to solve equations through problems prefaced with gripping phrases like “I found a stone but did not weigh it.” For example, one such problem continues: “After I weighed out six times its weight, added two gin [a unit of weight] and added a third of one seventh [of this new weight] multiplied by 24, I weighed it. The result was 1 ma-na (equal to 60 gin). What was the original weight of the stone?” Babylonian schoolboys had to know how to solve this kind of problem. Can you? (Find the answer below).
Mathematical puzzles have interested people throughout history, often with an educational slant. Alcuin of York’s Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young) of about 780 AD contains the famous river-crossing puzzle: a traveller reaches the bank of a river with a wolf, a goat, and a cabbage. There is a boat, but it can carry only the traveller and one animal or the cabbage. If left alone together, the goat will eat the cabbage and the wolf will eat the goat. However, the wolf hates cabbage. How does the traveller get his animals and his cabbage to the other side, uneaten, using the minimum number of trips across the river?
The French mathematician Édouard Lucas invented several ingenious puzzles, the most famous being the Tower of Brahma, published in 1883, which featured in an early episode of Dr Who. A temple contains three posts, on one of which is a pile of 64 golden discs with holes in their centres. The discs are all of different sizes, and they start in order of size with the largest at the bottom. Brahmin priests are moving the discs to another post, one at a time, but may never place a disc on top of a smaller one. When they complete the task, the world will end. Can they do so? If they can, what is the smallest number of moves required?
Over the years, the Prospect Enigma has included a large cast of bizarre characters and a lot of hidden mathematics. Farmer Suticle’s animals playing on a seesaw provide a chance to brush up on your algebra. The Number Monks of Wuntumenni’s wish to design a new prayer-patio leads to basic geometry and number theory. Catticus fatticus and Pussius fussius, felinoids of the planet Katzwiskas, illustrate principles of Diophantine equations.
I get inspiration for characters and situations from many sources: current affairs, famous movies, TV series, my own penchant for cats and pigs. Some I just make up. The Latin names for the felinoids were invented years ago by my younger son Chris, referring to our own cats. I have a liking for quintessentially English villages, which also make appearances in my Enigmas, such as Much Grumbling and Lower Aspirations. And I find parodies (along with silly names) irresistible, such as the bar in Mos Unleikly spaceport, where Stan Polo and Duke Slyhawker discuss rescuing the princess (revealed in a later column as Hava Banana) from the Clone Room on Venupiter, where Veracitors and Dissimulators respectively always tell the truth and always lie.
Without doubt, though, my favourite character is the adventurous archaeologist Colorado Smith, not forgetting his trusty sidekick Brunnhilde. Earthquakes, ravenous hordes, bottomless chasms—Smith brushes them aside, scarcely deigning to notice them. He is hopelessly sloppy, often forgetting to bring key items of information, but he has boundless confidence in his ability to find a way out of any predicament. He is so cool. So it is only fitting to end with his final adventure.
Colorado Smith and the last roulade
“Choose!” Smith looked at the array of food arranged before him, and knew he was in trouble. “Brunnhilde! You realise what this means?”
“No?” “We have been captured by the dessert nomads of Krizmazdinna!”
“Shouldn’t that be ‘desert’?”
“No! These evil fiends ply their captives with irresistibly tasty puddings, one of which is poisoned! If they leave that one to last, it is taken away and they are released, but if they eat it... no, it is too horrible to describe.”
“Be quiet, looter of sacred sites!” a bearded face snarled, as its owner brandished a large serving spoon.
“Well, how was I to know that your tribe would have the slightest interest in the fabled Lapis Lazuli Mines of Shangri-tra-la-la? And it was only a small idol that got smashed—”
“Shut up and choose!” Smith surveyed the feast. There were 12 desserts on a round cloth, starting with Christmas pudding at one o’clock and ending with a huge chocolate roulade at 12 o’clock.
The rules were simple. Begin by choosing a dessert and eating it. Then move seven desserts clockwise, and eat that. Then seven clockwise again, and so on. The gaps that appear are not included in the count, and you might have to go round the circle several times. When just one dessert remains, stop. The roulade contains the poison, so the aim is to leave that uneaten.
“Where do we begin, Smith?” asked Brunnhilde.
“The secret is recorded in my father’s notebook.”
“Ah! You have it with you, no doubt?”
“No, I dropped it into the great volcano Eyjapetlkrakajökull when we were escaping from those Icelaztonesian headhunters. But worry not: we have several seconds to work it out.”
Where should they begin?
