Tag Archives: logic

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

Pinocchio’s Hats

This problem in logic from Presh Talwalkar recalled an article I wrote a while ago but did not publish.  So I thought I would post it as part of the solution.

“Assume that both of the following sentences are true:

  1. Pinocchio always lies;
  2. Pinocchio says, “All my hats are green.”

We can conclude from these two sentences that:

  • (A) Pinocchio has at least one hat.
  • (B) Pinocchio has only one green hat.
  • (C) Pinocchio has no hats.
  • (D) Pinocchio has at least one green hat.
  • (E) Pinocchio has no green hats.”

Actually, the question is which, none or more, of statements (A) – (E) follow from the two sentences?

See Pinocchio’s Hats

Shared Spaces Puzzle

This is a nice puzzle from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008.  It is more a logic puzzle than a geometric one.

“In the diagram, each question mark represents one of six consecutive whole numbers. The sum of the numbers in the triangle is 39, the sum of those in the square is 46 and the sum of those in the circle is 85.  What are the six numbers?”

See the Shared Spaces Puzzle

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

Heron Suit Problem

Here is another logic problem from Ian Stewart.

  1. No cat that wears a heron suit is unsociable.
  2. No cat without a tail will play with a gorilla.
  3. Cats with whiskers always wear heron suits.
  4. No sociable cat has blunt claws.
  5. No cats have tails unless they have whiskers.

Therefore:

No cat with blunt claws will play with a gorilla.

Is the deduction logically correct?

I confess I don’t know what a heron suit is.  Google showed various garments with herons imprinted on the cloth, so maybe that is what it is.

See the Heron Suit Problem

Mystery of the Dancing Men

Manmohan Kaur took Arthur Conan Doyle’s popular 1903 Sherlock Holmes story “The Adventure of the Dancing Men” and used it in his math classes to illustrate the logic and mathematics involved in solving codes and ciphers.  I thought his idea might work as a puzzle.  It has been years since I read the story, so I had forgotten the decryption and found it quite doable from the setup provided by Kaur.  Here is his presentation, subject to further edits and reductions in size on my part.

“The original story has been shortened and simplified. Reference to England has been completely removed and some other superfluous information that distracts the reader instead of helping solve the mystery have been omitted. In the original story Elriges is the name of an inn but we have taken the liberty to use it loosely as the name of a town.

The pictures of all stick figure messages except the fourth are from the collection The Return of Sherlock Holmes. The original story has a typographical error that throws off the decryption scheme. To remove this (intentional or unintentional) error, the fourth figure has been taken from Trap and Washington’s Introduction to Cryptogra­phy with Coding Theory. The fourth message is meant to have a different handwriting, so this serves our purposes well.

Condensed Story

Hilton Cubitt of Elriges visits you and gives you a paper with the following mysterious sequence of stick figures that he found lying on the sun-dial in his mansion.

Message 1:

Cubitt explains that he recently married a Chicago woman named Elsie Patrick. Before the wedding, she had asked him never to ask about her past, as she had had some “very disagreeable associations” in her life, although she said that there was nothing that she was personally ashamed of. Their mar­riage had been a happy one until the messages began to arrive, first mailed from Chicago and then appearing in the garden of his mansion.

The messages had made Elsie very afraid but she did not explain the reasons for her fear, and Cubitt insisted on honoring his promise not to ask about Elsie’s life in Chicago. You look at the figures closely to understand them a little better and notice that some of the figures are holding flags. What could the flags mean? Perhaps the end of words?

The next morning Cubitt finds “a fresh crop of dancing men drawn in chalk upon the black wooden door of the tool-house”:

Message 2:

Two mornings later, “a fresh inscription had appeared”:

Message 3:

Three days later, “a message was left scrawled upon paper, and placed under a pebble upon the sun-dial”:

Message 4:

Cubitt gives copies of all these messages to you. Your task is to help him understand what is going on. You call your friend in the Chicago Police Department and ask her to find background information on Elsie Patrick. You learn that Elsie is the daughter of a Chicago crime boss, and was engaged to Abe Slaney, who worked for her dad, and that she had fled to escape her old life.

You examine all the occurrences of the dancing figures. Message 4 is in a different handwriting, so you guess that it is from a different person, most likely, Elsie, while messages 1, 2 and 3 are from the unknown person (the criminal). You spend the next two days trying to make some sense of the stick figures. You are now sure that the flags on some of the figures indicate the end of words. You also know that a simple substitution cipher is being used for the encryption, and that frequency analysis is the way to solve these ciphers.

Three days later, another message appears.

Message 5:

This message causes you to fear that the Cubitts are in immediate danger. You rush to Elriges and find Cubitt dead of a bullet to the heart and his wife gravely wounded from a gunshot to the head. What do the messages say?

Inspector Martin of the Norfolk Constabulary believes that it is a murder-suicide attempt; Elsie is the prime suspect. But you, after noting some inconsistencies in that theory, know that there is a third person involved. How will you prove to Inspector Martin that a third person is involved?”

See The Mystery of the Dancing Men

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

Puzzle of the Purloined Papers

Ian Stewart has a nice logic problem in his Casebook of Mathematical Mysteries, which includes a pastiche of Sherlock Holmes in the form of Herlock Soames and Dr. Watsup, along with brother Spycraft and nemesis Dr. Mogiarty.

“An important document was accidentally mislaid, and then stolen,” Spycraft said. “It is essential to the security of the British Empire that it be recovered without delay. If it gets into the hands of our enemies, careers will be ruined and parts of the Empire may fall. Fortunately, a local constable caught a glimpse of the thief, enough to narrow it down to precisely one of four men.”

“Petty thieves?”

“No, all four are gentlemen of high repute. Admiral Arbuthnot, Bishop Burlington, Captain Charlesworth, and Doctor Dashingham.”

Soames sat bolt upright. “Mogiarty has a hand in this, then.”

Not following his reasoning, I asked him to explain.

“All four are spies, Watsup. Working for Mogiarty.”

“Then … Spycraft must be engaged in counter-espionage!” I cried.

“Yes.” He glanced at his brother. “But you did not hear that from me.”

“Have these traitors been questioned?” I asked.

Spycraft handed me a dossier, and I read it aloud for Soames’s benefit. “Under interrogation Arbuthnot said ‘Burlington did it.’ Burlington said ‘Arbuthnot is lying.’ Charlesworth said ‘It was not I.’ Dashingham said ‘Arbuthnot did it.’ That is all.”

“Not quite all. We know from another source that exactly one of them was telling the truth.”

“You have an informer in Mogiarty’s inner circle, Spycraft?”

“We had an informer, Hemlock. He was garrotted with his own necktie before he could tell us the actual name. Very sad—it was an Old Etonian tie, totally ruined. However, all is not lost. If we can deduce who was the thief, we can obtain a search warrant and recover the document. All four men are being watched; they will have no opportunity to pass the document to Mogiarty. But our hands are tied; we must stick to the letter of the law. Moreover, if we raid the wrong premises, Mogiarty’s lawyers will publicise the mistake and cause irreparable damage.”

Which man was the thief?

See the Purloined Papers Puzzle.

Rock, Paper, Scissors Problem

Here is another problem from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).

“Wiggins grinned at me. ‘You’ve not played Rock Paper Scissors before, Doctor?’

‘Doesn’t ring a bell,’ I told him.

‘Two of you randomly pick one of the three, and shout your choice simultaneously. There are hand gestures, too. If you both get the same, it’s a draw. Otherwise, scissors beats paper, paper beats rock, and rock beats scissors.’

‘So it’s a way of settling an argument,’ I suggested.

‘You were brought up wrong, Doctor,’ Wiggins said gravely. ‘Look, try it this way. I played a series of ten games with Alice earlier. I picked scissors six times, rock three times, and paper once. She picked scissors four times, rock twice, and paper four times. None of our games were drawn.’ He glanced at Holmes, who nodded. ‘So then, Doctor. What was the overall score for the series?’ ”

See the Rock Paper Scissors Problem

(Update 7/29/2021)  This problem in a different guise was presented by Futility Closet (7/28/2021) and attributed to Yoshinao Katagiri in Nobuyuki Yoshigahara’s Puzzles 101: A Puzzlemaster’s Challenge, 2004.

Do I Avoid Kangaroos?

This is a fun logic puzzle from one of Ian Stewart’s many math collections.  I discovered that the problem actually is basically one of Lewis Carroll’s examples from an 1896 book:

  1. The only animals in this house are cats.
  2. Every animal that loves to gaze at the moon is suitable for a pet.
  3. When I detest an animal, I avoid it.
  4. No animals are meat-eaters, unless they prowl by night.
  5. No cat fails to kill mice.
  6. No Animals ever take to me, except those in this house.
  7. Kangaroos are not suitable for pets.
  8. Only meat-eaters kill mice.
  9. I detest animals that do not take to me.
  10. Animals that prowl at night love to gaze at the moon.

If all these statements are correct, do I avoid kangaroos, or not?

See Do I Avoid Kangaroos?