Tag Archives: Alex Bellos

The Maths of Lviv

Unfortunately Ukraine has receded from our attention under the threat from our own anti-democratic forces, but this Monday Puzzle from Alex Bellos in March is a timely reminder of the mathematical significance of that country.

“Like many of you I’ve hardly been able to think about anything else these past ten days apart from the war in Ukraine. So today’s puzzles are a celebration of Lviv, Ukraine’s western city, which played an important role in the history of 20th century mathematics. During the 1930s, a remarkable group of scholars came up with new ideas, methods and theorems that helped shape the subject for decades.

The Lwów school of mathematics – at that time, the city was in Poland – was a closely-knit circle of Polish mathematicians, including Stefan Banach, Stanisław Ulam and Hugo Steinhaus, who made important contributions to areas including set-theory, topology and analysis. …

Of the many ideas introduced by the Lwów school, one of the best known is the “ham sandwich theorem,” posed by Steinhaus and solved by Banach using a result of Ulam’s. It states that it is possible to slice a ham sandwich in two with a single slice that cuts each slice of bread and the ham into two equal sizes, whatever the size and positions of the bread and the ham.

Today’s puzzles are also about dividing food. The first is from Hugo Steinhaus’ One Hundred Problems in Elementary Mathematics, published in 1938. The second uses a method involved in the proof of the ham sandwich theorem.

  1. Three friends each contribute £4 to buy a £12 ham. The first friend divides it into three parts, asserting the weights are equal. The second friend, distrustful of the first, reweighs the pieces and judges them to be worth £3, £4 and £5. The third, distrustful of them both, weighs the ham on their own scales, getting another result. If each friend insists that their weighings are correct, how can they share the pieces (without cutting them anew) in such a way that each of them would have to admit they got at least £4 of ham?_
  2. Ten plain and 14 seeded rolls are randomly arranged in a circle, equidistantly spaced, as below. Show that using a straight line it is possible to divide the circle into two halves such that there are an equal number of plain and seeded rolls on either side of the line.

Show there is always a diameter that cuts the circle into two batches of 12 rolls with an equal number of plain and seeded.

Question 2 is adapted from Mathematical Puzzles by Peter Winkler, who gives as a reference Alon and West, The Borsuk-Ulam Theorem and bisection of necklaces, Proceedings of the American Mathematical Society 98 (1986).

See The Maths of Lviv

Date Night

This is a fairly straight-forward logic puzzle from Alex Bellos’s Monday Puzzle in The Guardian.

“When it comes to the world of mathematical puzzles, Hungary is a superpower. Not just because of the Rubik’s cube, the iconic toy invented by Ernő Rubik in 1974, but also because of its long history of maths outreach.

In 1894, Hungary staged the world’s first maths competition for teenagers, four decades before one was held anywhere else. 1894 also saw the launch of KöMaL, a Hungarian maths journal for secondary school pupils full of problems and tips on how to solve them. Both the competition and the journal have been running continuously since then, with only brief hiatuses during the two world wars.

This emphasis on developing young talent means that Hungarians are always coming up with puzzles designed to stimulate a love of mathematics. (It also explains why Hungary arguably produces, per capita, more top mathematicians than any other country.)

I asked Béla Bajnok, a Hungarian who is now director of American Mathematics Competitions, a series of competitions involving 300,000 students in the US, whether he knew of any puzzles that originated in Hungary. The first thing he said that came to mind was the ‘3-D logic puzzle’, a type of logic puzzle in which you work out the solution in a three dimensional box, rather than (as is the case with the standard version) in a two-dimensional grid. He said he had never seen this type of puzzle outside Hungary.

Below are two examples he created. You could solve these using an extended two dimensional grid. It’s more in the spirit of the question, however, to draw a three-dimensional one, like you are looking at three sides of a Rubik’s Cube.

Date night

Andy, Bill, Chris, and Daniel are out tonight with their dates, Emily, Fran, Gina, and Huong. We have the following information.

  1. Andy will go to the opera
  2. Bill will spend the evening with Emily,
  3. Chris would not want to go out with Gina,
  4. Fran will see a movie
  5. Gina will attend a workshop.

We also know that one couple will see an art exhibit. Who will go out with whom, and what will they do?

See Date Night

Numbers in New Guinea

This puzzle from Alex Bellos follows the themes in his new book, The Language Lover’s Puzzle Book, which, among other things, looks at number systems in different languages.  (See also his Numberphile video.)

“Today is the International Day of the World’s Indigenous People, which aims to raise awareness of issues concerning indigenous communities. Such as, for example, the survival of their languages. According to the Endangered Languages Project, more than 40 per cent of the world’s 7,000 languages are at risk of extinction.

Among the fantastic diversity of the world’s languages is a diversity in counting systems. The following puzzle concerns the number words of Ngkolmpu, a language spoken by about 100 people in New Guinea. (They live in the border area between the Indonesian province of Papua and the country of Papua New Guinea.)

Ngkolmpu-zzle

Here is a list of the first ten cube numbers (i.e. 13, 23, 33, …, 103):

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Below are the same ten numbers when expressed in Ngkolmpu, but listed in random order. Can you match the correct number to the correct expressions?

eser tarumpao yuow ptae eser traowo eser

eser traowo yuow

naempr

naempr ptae eser traowo eser

naempr tarumpao yuow ptae yuow traowo naempr

naempr traowo yempoka

tarumpao

yempoka tarumpao yempoka ptae naempr traowo yempoka

yuow ptae yempoka traowo tampui

yuow tarumpao yempoka ptae naempr traowo yuow

Here’s a hint: this is an arithmetical puzzle as well as a linguistic one. Ngkolmpu does not have a base ten system like English does. In other words, it doesn’t count in tens, hundreds and thousands. Beyond its different base, however, it behaves very regularly.

This puzzle originally appeared in the 2021 UK Linguistics Olympiad, a national competition for schoolchildren that aims to encourage an interest in languages. It was written by Simi Hellsten, a two-time gold medallist at the International Olympiad of Linguistics, who is currently reading maths at Oxford University.”

See Numbers in New Guinea

Snooker Puzzle

This is a nice puzzle from Alex Bellos’s Monday Puzzle column in the Guardian.

“My cultural highlight of recent weeks has been the brilliant BBC documentary Gods of Snooker, about the time in the 1980s when the sport was a national obsession. Today’s puzzle describes a shot to malfunction the Romford Robot … and put the Whirlwind … in a spin.

Baize theorem

A square snooker table has three corner pockets, as [shown]. A ball is placed at the remaining corner (bottom left). Show that there is no way you can hit the ball so that it returns to its starting position.

The arrows represent one possible shot and how it would rebound around the table.

The table is a mathematical one, which means friction, damping, spin and napping do not exist. In other words, when the ball is hit, it moves in a straight line. The ball changes direction when it bounces off a cushion, with the outgoing angle equal to the incoming angle. The ball and the pockets are infinitely small (i.e. are points), and the ball does not lose momentum, so that its path can include any number of cushion bounces.

Thanks to Dr Pierre Chardaire, associate professor of computing science at the University of East Anglia, who devised today’s puzzle.”

See the Snooker Puzzle

The Squirrel Puzzle

For a change of pace, here is an early puzzle from Alex Bellos in The Guardian.

“Happy New Year guzzlers!  Today’s first problem concerns squirrels. Have a nibble—it’s not too hard a nut to crack.

The Squirrel King has buried the Golden Acorn beneath one of the squares in this 6x6 grid. Three squirrels—Black, Grey and Red—are each standing on a square in the grid, as illustrated.

(Note: for the purposes of today, squirrels can speak, hear, read, count and are perfect logicians. They can also move in any direction horizontally and vertically, not just the direction these cartoons are facing. They all can see where each other is standing, and the cells in the grid are to be considered squares.)

The Squirrel King hands each squirrel a card, on which a number is written. The squirrels can read only the number on their own card. The King tells them: ‘Each card has a different number on it, and your card tells you the number of steps you are from the square with the Golden Acorn. Moving one square horizontally or vertically along the grid counts as a single step.’ (So if the acorn was under Black, Black’s card would say 0, Grey’s would say 4, and Red’s 5. Also, the number of steps given means the shortest possible number of steps from each squirrel to the acorn.)

The King asks them: ‘Do you know the square where the Golden Acorn is buried?’ They all reply ‘no!’ at once.

Red then says: ‘Now I know!’

Where is the Golden Acorn buried? …”

See the Squirrel Puzzle

Three Counting Puzzles

Here are three counting puzzles from Alex Bellos’s book, Can You Solve My Problems?  Bellos recalls the famous legend of the young Gauss in the 19th century who summed up the whole numbers from 1 to 100 by finding a pattern that would simplify the work.  Bellos also mentioned that Alcuin some thousand years earlier had discovered a similar, but different, pattern to sum up the numbers.  In presenting these three problems he said, “The lesson … is this: If you’re asked to add up a whole bunch of numbers, don’t undertake the challenge literally.  Look for the pattern and use it to your advantage.”

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Ant Problem

This is one of Alex Bellos’s Monday Puzzles in the Guardian. I basically found the same solution as Bellos and his commenters, but wrote it up with what I thought were more explanatory graphics. The idea is that there is a bunch of ants on a stick who all walk a the same speed of 1 centimeter per second. When an ant runs into another ant, they both turn around and go the opposite direction. “So here is the puzzle: Which ant is the last to fall off the stick? And how long will it be before he or she does fall off?”  See the Ant Problem.