Existence Proofs

Here is a seemingly simple problem from Futility Closet.

“A quickie from Peter Winkler’s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?”

Technically we might rephrase this as, can we inscribe a flat map of West Virginia in a square, since the boundary of most states is probably not differentiable everywhere, that is, has a tangent everywhere.

But the real significance of the problem is that it is an example of an “existence proof”, which in mathematics refers to a proof that asserts the existence of a solution to a problem, but does not (or cannot) produce the solution itself.  These proofs are second in delight only to the “impossible proofs” which prove that something is impossible, such as trisecting an angle solely with ruler and compass.

Here is another classic example (whose origin I don’t recall).  Consider the temperatures of the earth around the equator.  At any given instant of time there must be at least two antipodal points that have the same temperature.  (Antipodal points are the opposite ends of a diameter through the center of the earth.)

See Existence Proofs (revised)

(Update 10/2/2021) I fixed a minor typo: “tail” should have been “head”

Post Views: 1,573

Twisting Beam Problem

Here is a slightly different kind of problem from the Polish Mathematical Olympiads.

“106. A beam of length a is suspended horizontally by its ends by means of two parallel ropes of lengths b.  We twist the beam through an angle φ about the vertical axis passing through the centre of the beam.  How far will the beam rise?”

Answer.

See the Twisting Beam Problem for solutions.

Post Views: 1,570

Topple Blocks Puzzle

This is another imaginative puzzle from MEI’s MathsMonday.  I didn’t know what topple blocks were at first, but a previous MathsMonday puzzle defined them as shown in the first frame.  Apparently they derive from the game of Jenga created by Leslie Scott and launched in 1983.

See the Topple Blocks Puzzle

Post Views: 1,417

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

Post Views: 1,408

Polynomial Division Problem

Here is a challenging problem from the Polish Mathematical Olympiads published in 1960.

“22. Prove that the polynomial

x44 + x33 + x22 + x11 + 1

is divisible by the polynomial

x4 + x3 + x2 + x + 1.”

See the Polynomial Division Problem

(Update 8/23/2021)  The idea expressed in this post that mathematicians are “lazy” and seek short-cuts to solving questions and problems, as I did in this one, was recently the subject of a Numberphile post by Marcus du Sautoy: “Mathematics is all about SHORTCUTS“.

Post Views: 2,122

Walking Banker Problem

Here is another Brainteaser from the Quantum magazine.

“Mr. R. A. Scall, president of the Pyramid Bank, lives in a suburb rather far from his office. Every weekday a car from the bank comes to his house, always at the same time, so that he arrives at work precisely when the bank opens. One morning his driver called very early to tell him he would probably be late because of mechanical problems. So Mr. Scall left home one hour early and started walking to his office. The driver managed to fix the car quickly, however, and left the garage on time. He met the banker on the road and brought him to the bank. They arrived 20 minutes earlier than usual. How much time did Mr. Scall walk? (The car’s speed is constant, and the time needed to turn around is zero.) (I. Sharygin)”

I struggled with some of the ambiguities in the problem and made my own assumptions.   But it turned out there was a reason they were ambiguous.

Answer.

See the Walking Banker Problem for solutions.

Post Views: 1,542

Triangle Quadrangle Puzzle

This is another simple problem from Five Hundred Mathematical Challenges:

“Problem 57.  Let X be any point between B and C on the side BC of the convex quadrilateral ABCD (as in the Figure).  A line is drawn through B parallel to AX and another line is drawn through C parallel to DX.  These two lines intersect at P.  Prove that the area of the triangle APD is equal to the area of the quadrilateral ABCD.”

See the Triangle Quadrangle Puzzle

Post Views: 5,249

The Pearl Necklace Problem

This problem comes from the Scottish Mathematical Council (SMC) Senior Mathematical Challenge of 2008:

“S2. In Tiffany’s, a world famous jewellery store, there is a string necklace of 33 pearls. The middle one is the largest and most valuable. The pearls are arranged so that starting from one end, each pearl is worth $100 more than the preceding one, up to [and including] the middle one; and starting from the other end, each pearl is worth $150 more than the preceding one, up to [and including] the middle one. If the total value of the necklace is $65,000 what is the value of the largest pearl?”

I included the words in brackets to erase any ambiguity.

Answer.

See the Pearl Necklace Problem for solutions.

Post Views: 4,538

Fashion Puzzle

Again we have a puzzle from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).

“On one occasion, Holmes and I were asked to, solve the robbery of a number of dresses from the workshop of a recently deceased ladies’ tailor to the upper echelons of society. Holmes took a short look at the particulars of the case, and sent them all back to the gown-maker’s son with a scribbled note to the effect that it could only be one particular seamstress, with the help of her husband.

However, glancing through my observations some period later, I observed certain facts about the robbery which led me to an interesting little exercise. The stock at the workshop had been very recently valued at the princely sum of £1,800, and when examined after the theft, comprised of precisely 100 completed dresses in a range of styles, but of equal valuation. However, there was no remaining record of how many dresses had been there beforehand. The son did recall his father stating, of the valuation, that if he’d had thirty. dresses more, then a valuation of £1,800 would have meant £3 less per dress.

Are you able to calculate how many dresses were stolen?”

Answer.

See the Fashion Puzzle for solutions.

Post Views: 1,547

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?

Answer.

See the Purloined Papers Puzzle for solutions.

Post Views: 1,533
  1. Pages:
  3. 1
  4. ...
  5. 20
  6. 21
  7. 22
  8. 23
  9. 24
  10. 25
  11. 26
  12. ...
  13. 46