Quick Thinking

Some “ridiculous questions” from Martin Gardner:

1. A convex regular polyhedron can stand stably on any face, because its center of gravity is at the center. It’s easy to construct an irregular polyhedron that’s unstable on certain faces, so that it topples over. Is it possible to make a model of an irregular polyhedron that’s unstable on every face?

2. The center of a regular tetrahedron lies in the same plane with any two of its corner points. Is this also true of all irregular tetrahedrons?

3. An equilateral triangle and a regular hexagon have perimeters of the same length. If the area of the triangle is 2 square units, what is the area of the hexagon?

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1. No. If this were possible then a perpetual motion machine could be built — on a horizontal plane the object would be constantly moving. 2. Yes. Any three points lie in the same plane. 3. Three square units: (Martin Gardner, “Ridiculous Questions,” Math Horizons 4:2 [November 1996], 24-25.)

October 24, 2015November 1, 2015 | Puzzles · Science & Math

What’s the Angle?

what's the angle puzzle

AB = XY. Find z°.

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Construct equilateral triangle ABW. Now: ∠WAY = 40° (because each interior angle of ABW is 60° and the interior angles of triangle XAY must total 180°) p = p (because triangle ATY is isosceles) q = q (because AB = AW = XY = p + q) ∠AWY = 40° (because triangles ATX and YTW are similar) r = r (because triangle AYW is isosceles) This shows that ABWY is a kite, a quadrilateral with two pairs of equal-length sides, each pair adjacent. The diagonals of a kite are perpendicular, so the triangle that contains angle z also contains interior angles measuring 90° and 80°. Angle z is 10°. (Thanks, Derek.)

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October 22, 2015November 1, 2015 | Puzzles · Science & Math

More Fun

If the proportion of blonds among blue-eyed people is greater than among the population as a whole, is it also true that the proportion of blue-eyed people among blonds is greater than among the population as a whole?

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Yes. Let a be the number of blonds with blue eyes, b be the number of all blonds, c be the number of all blue-eyed people, and n be the whole population. Then by the information we’re given, a/c > b/n, so an > bc and a/b > c/n. (By Anatoly Savin)

October 19, 2015November 12, 2023 | Puzzles

Earnings Report

My employer has nine workers. The nine of us want to determine what our average salary is, but none of us wants to divulge his own salary. Can we find the average without doing so?

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Yes. I privately pick some large random number — say, 645,743 — add my own salary to it, and whisper the total to Worker #2. He adds his own salary to that sum and whispers the new total to Worker #3, and so on. When the last worker has added his own salary, he whispers the final amount back to me. I subtract the random number that I’d chosen and divide the remainder by 9. Now we know the average salary of the nine workers, but none of us knows anything about what the others make. 11/03/2015 UPDATE: The solution above is not quite optimal because the workers can still make some inferences about their coworkers’ maximum salaries. For example, if I give Worker #2 the number 701,229, he knows that my salary cannot be higher than that. This can be improved by allowing the random number to be negative — in that case no inferences are possible, and the math still works. (Thanks, Daniel and Jose.)

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October 17, 2015November 3, 2015 | Puzzles

Black and White

von meyenfeldt chess puzzle

A poser by F.H. von Meyenfeldt, 1967. What move must Black play to enable a forced mate in two by White?

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… Bh3 enables 1. Rh1, and Black can’t stop 2. Bg7#. From Stephen Addison, The Book of Extraordinary Chess Problems, 1989. (10/12/2015 This seems to be cooked. After … Bg4, … Bd7, or Bc8, White can withdraw his bishop along c1-h6 and mate on the long diagonal. Thanks, Malcolm.)

October 11, 2015October 18, 2015 | Puzzles

Remainders

Some geometric legerdemain by Argentine magician Norberto Jansenson. (Thanks, Ron.)

October 11, 2015October 10, 2015 | Entertainment · Puzzles · Science & Math

Shifting Gears

A puzzle by French puzzle maven Pierre Berloquin:

Timothy rides a bicycle on a road that has four parts of equal length.

The first fourth is level, and he pedals at 10 kph.

The second fourth is uphill, and he pedals at 5 kph.

The third fourth is downhill, and he rides at 30 kph.

The fourth fourth is level again, but he has the wind at his back, so he goes 15 kph.

What is his average speed?

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Let L be the length in kilometers of each fourth of the road. Then Timothy covers the first fourth in L/10 hours, the second fourth in L/5 hours, the third fourth in L/30 hours, and the fourth fourth in L/15 hours. The total time is L/10 + L/5 + L/30 + L/15 = 2L/5. Since speed equals distance divided by time, we can get Timothy’s average speed by dividing the length of the whole road, 4L, by the total time: $4L \div \displaystyle\frac{2L}{5} = 10\: \mathrm{kph}$

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October 8, 2015October 18, 2015 | Puzzles

Cruel and Unusual

A king is angry at two mathematicians, so he decrees the following punishment. The mathematicians will be imprisoned in towers at opposite ends of the kingdom. Each morning, a guard at each tower will flip a coin and show the result to his prisoner. Each prisoner must then guess the result of the coin flip at the other tower. If at least one of the two guesses is correct, they will live another day. But as soon as both guesses are incorrect, they will be executed.

On the way out of the throne room, the mathematicians manage to confer briefly, and they come up with a plan that will spare them indefinitely. What is it?

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They agree that Mathematician A will always guess the result of his own flip, and Mathematician B will always guess the opposite of his own flip. This guarantees that at least one guess will always be correct. (From Max Schireson’s blog.) (Thanks, Arianna.)

October 1, 2015October 11, 2015 | Puzzles

Wanted

Items requested in the 2000 Baylor College Linguistic Scavenger Hunt:

the word for “cheese” in Estonian
the longest word in English that uses no letter more than once
a nine-letter English word that has only one syllable
the sound that a dog makes in Swedish
the regional word for “drinking fountain” that’s used in Wisconsin
the language that Jesus spoke
the American equivalent of the British word “ex-directory”
five words that are legal plays in Scrabble and that have only two letters, one of which is “x”
the motto of the Klingon Language Institute
identity of the person who said, “England and America are two countries divided by a common language.”

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juust uncopyrightable [also dermatoglyphics] strengths or screeched vov vov bubbler Aramaic unlisted (phone number) ax, ex, xi, ox, xu “Language opens worlds.” George Bernard Shaw There’s more in the Winter 2000 issue of Verbatim (page 12).