Bellos Can You Solve My Problems?

A casebook of ingenious, perplexing and totally satisfying puzzles
Main
ISBN: 978-1-78335-116-9
Verlag: Guardian Faber Publishing
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 288 Seiten

Are you smarter than a Singaporean ten-year-old? Can you beat Sherlock Holmes? If you think the answer is yes - I challenge you to solve my problems. Here are 125 of the world's best brainteasers from the last two millennia, taking us from ancient China to medieval Europe, Victorian England to modern-day Japan, with stories of espionage, mathematical breakthroughs and puzzling rivalries along the way. Pit your wits against logic puzzles and kinship riddles, pangrams and river-crossing conundrums. Some solutions rely on a touch of cunning, others call for creativity, others need mercilessly logical thought. Some can only be solved be 2 per cent of the population. All are guaranteed to sharpen your mind. Let's get puzzling!

Alex Bellos is a grandmaster of the puzzling world, brilliant on all things cryptic. His bestselling, award-winning books include Alex's Adventures in Numberland, Alex Through the Looking-Glass and Can You Solve My Problems?, and have been translated into more than twenty languages. He is also the coauthor of two mathematical colouring books and the children's series Football School. His YouTube videos have been seen by more than twenty million people, and he writes a popular puzzle blog for the Guardian. @alexbellos

Autoren/Hrsg.

Bellos, Alex

Leseproben

Cabbages, Cheating Husbands and a Zebra

LOGIC PROBLEMS

Logic. It’s a logical place to start: logical deduction is the ground rule of all mathematical puzzles. Indeed, logic is the foundation of all mathematics. In the nomenclature of puzzledom, however, ‘logic problems’ are brainteasers that employ deductive reasoning alone – shunning, for example, any type of arithmetical calculation, algebraic manipulation, or sketching of shapes on the backs of envelopes. They are the most accessible type of mathematical conundrum because they require no technical knowledge, and the questions easily lend themselves to humorous phrasing. But, as we shall see, they are not always the easiest to solve, since they twist our brains in unfamiliar ways.

Which they have been doing since at least the time of Charlemagne, King of the Franks.

*

In 799 CE, Charlemagne, who ruled over much of Western Europe, received a letter from his old teacher, Alcuin: ‘I have sent you’, it read, ‘some arithmetical curiosities to amuse you.’

Alcuin was the greatest scholar of his era. He grew up in York, attending and then running the city’s cathedral school, the best educational establishment in the country. The Englishman’s reputation reached Charlemagne. The king persuaded him to run his palace school in Aachen, where Alcuin founded a large library and went on to reform education across the Carolingian empire. Alcuin eventually left Charlemagne’s court to become Abbot of Tours, which is when he wrote the above letter to his former boss.

Alcuin is credited by some with inventing joined-up writing so he and his many scribes could write faster. Some believe that he was also the first person to use a symbol – a diagonal squiggle – as punctuation for a question. It is wonderfully appropriate that the predominant early figure in the history of puzzles was also a progenitor of the question mark.

The physical document that Alcuin was referring to in his letter to Charlemagne no longer exists, but historians believe that it was a list of fifty or so problems called Propositiones ad Acuendos Juvenes, or Problems to Sharpen the Young, of which the earliest surviving manuscript dates from a century later. Who else, they argue, could have written it but Alcuin, the foremost teacher of his day?

Propositiones is a remarkable document. It is the largest cache of puzzles from medieval times, as well as the first Latin text that contains original mathematical material. (The Romans may have built roads, aqueducts, public baths and sanitation systems, but they never did any mathematics.) It begins in thigh-slapping tone:

A swallow invites a snail for lunch a league away. If the snail travels an inch a day, how long will it take him?

The answer is 246 years and 210 days. He would have died more than two centuries before he got there.

Another one asks:

A certain man met some students and asked them: ‘How many of you are there in your school?’ One of the students replied: ‘I don’t want to tell you directly but I’ll tell you how to work it out. You double the number of students, then triple that number, then divide that number into four parts. If you add me to one of the quarters, there will be 100.’ How many students are there in the school?

Pesky kids! I’ll leave this one for you to solve on your own.

Alcuin’s whimsical phrasing was groundbreaking. It was the first time humour had been used to pique students’ interest in arithmetic. Yet Propositiones was important not just because of its stylistic innovations, but also because it included several new types of problem. Some of them required deductive reasoning but no calculation. The best known of Alcuin’s puzzles is arguably the most famous mathematical riddle of all time.

1
WOLF, GOAT AND CABBAGES

A man arrives at a riverbank with a wolf, a goat and a bunch of cabbages. He needs to cross the river, but the one boat available can carry only him and a single item at the same time. He cannot leave the wolf alone with the goat or the goat alone with the cabbages, since in both cases the former will eat the latter.

How does he cross the river in the shortest number of crossings?

It’s a blinder for two reasons. For a start, the scene is comical. You’ve spent all morning trudging down a dirt path, desperately trying to keep the wolf away from the goat, and the goat away from the cabbages. Now your day just got worse: you have to cross a river in a stupidly tiny boat. Yet what I find most amusing and interesting about the scenario is its solution, which forces our hero to behave in a way you would not intuitively expect.

Have a go. All five-year-old children can solve it, declared one thirteenth-century text. No pressure.

Or follow the reasoning with me.

Let’s place the traveller on the left bank. He begins with three items and can only take one of them in the boat. If he takes the wolf, the goat will be left with the cabbages and eat them. If he takes the cabbages, the wolf will eat the goat. By a process of elimination, the only item he can take on the first crossing is the goat, since wolves don’t eat cabbages. He delivers the goat to the right bank and returns for the next item.

Now he has a choice of wolf or cabbages. Let’s say he decides to take the cabbages. He crosses the river for the third time. When he reaches the right bank he cannot leave the cabbages with the goat, so what does he do? He makes no progress if he returns with the cabbages, since he only just arrived with them, so he must return with the goat. This step is the counter-intuitive one: in order for the traveller to get everything across he needs to take something across, back, and across again.

Back on the left bank, after four crossings, with the wolf and the goat, the traveller chains the goat and departs for his fifth traverse with the wolf. On the right bank the wolf remains uninterested in the cabbages. All that is left is one trip back to pick up the bearded bovid, and our chap is done, in seven crossings.

(There’s a second, equivalent, solution: if he took the wolf on the second crossing, the same logic follows and he also finishes the job in seven trips.)

*

Propositiones also contains other river-crossing puzzles, such as this one, which sounds like the plot of a bedroom farce.

2
THREE FRIENDS AND THEIR SISTERS

So there were three of us men who each had a sister, and we all had to get across a river. But each of us lusted after one of the others’ sisters. On coming to the river all we found was a little ferry boat that could only carry two people at a time. Say if you can how we crossed the river so that none of our sisters was dishonoured by being alone in a boat with a man who was not her brother.

You can interpret this problem in two ways, since Alcuin’s phrasing is ambiguous. What’s not in dispute is that there are three couples, each consisting of a brother and a sister, who must all cross the river, and all that they have at their disposal is a two-person boat. But there could be either of two restrictions: [1] That a boat can never contain a woman and a man who are not related. In this case the entire party can reach the other side in nine crossings. [2] That a woman is forbidden from being in the boat unaccompanied by her brother when the boat is dropping off or collecting passengers at a bank where there are other men. The second scenario, I think, is more in the spirit of the question, and the mission requires eleven crossings. Try to find both solutions.

*

River-crossing puzzles have delighted children and adults for more than a thousand years. As they have spread across the world, they have changed to reflect local concerns. In Algeria, the wolf, goat and cabbages are a jackal, a goat and a bundle of hay; in Liberia they are a cheetah, a fowl and some rice; and in Zanzibar they are a leopard, a goat and some leaves. The puzzle of the three friends and their sisters has also evolved throughout the ages: the lecherous men soon became jealous husbands forbidding their wives to travel in the boat with another man. In one thirteenth-century retelling the couples have names: Bertoldus and Berta, Gherardus and Greta, and Rolandus and Rosa. The solution is presented as two hexameters. If you can read Latin, look away now:

Binae, sola, duae, mulier, duo, vir mulierque,

Bini, sola, duae, solus, vir cum muliere.

By the seventeenth century the couples were masters and valets. Each master forbade his valet to travel with another master in case that master murdered him. The social warfare was reversed in the...