Quickie

If I roll three dice and multiply the three resulting numbers together, what is the probability that the product will be odd?

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The product can be odd only if all three factors are odd: $\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$ (Thanks, Nick.) 02/01/2021 UPDATE: From reader Paul-Georg Becker: Another dice quickie: I have a set of n dice, but unfortunately only one of them is fair. If I roll all dice and add the n resulting numbers together, what is the probability that the product will be odd? Answer: For natural n let S_n be the sum of the resulting numbers x₁, …, x_n. We assume that x_n is the number shown by the fair die. Then we have P(S_n even) = P(x_n + S_n-1 even) = P(S_n-1 even) P(x_n even) + P(S_n-1 odd) P(x_n odd) = P(S_n-1 even) /2 + P(S_n-1 odd) / 2 = (P(S_n-1 even) + P(S_n-1 odd)) / 2 = 1/2 It does not even matter what the result for S_n-1 is. (Thanks, Paul-Georg.)

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January 27, 2021February 1, 2021 | Puzzles

Disappearing Act

https://commons.wikimedia.org/wiki/File:Anatomy_of_a_Sunset-2.jpg — Image: Wikimedia Commons

A puzzle by Joseph Horton, from MIT Technology Review, January-February 1999:

If the sun takes two minutes to set, what angle does it subtend from Earth?

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Half a degree. Earth rotates through 360 degrees every 24 hours, or 1 degree every 4 minutes. 01/27/2021 UPDATE: This is oversimplified — time to set depends on time of year (declination) and the observer’s latitude. The Review posted a correction in the May/June issue: “Peter Blicher notes that we are erroneously assuming the sun sets at a 90 degree angle to the horizontal.” Thanks, Chris and Martin.

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January 25, 2021January 27, 2021 | Puzzles

Black and White

dobrusky chess problem

By J. Dobrusky. White to mate in two moves.

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1. Qe8! and there’s no stopping a fatal rook check next move. From Benjamin Glover Laws, The Two-Move Chess Problem, 1890. 01/19/2021 UPDATE: There’s a pretty variation I’d overlooked: 1. Qe8 Kb5 2. Nxc3#. So in that case the knight delivers mate.

January 16, 2021January 19, 2021 | Puzzles

“Problem for Poultry Farmers”

Eureka, the esteemed journal of recreational mathematics published at Cambridge University, has placed its archives online under a Creative Commons license. Here’s a problem from May 1940:

The chicken was twice as old when when the day before yesterday was to-morrow to-day was as far from Sunday as to-day will be when the day after to-morrow is yesterday as it was when when to-morrow will be to-day when the day before yesterday is to-morrow yesterday will be as far from Thursday as yesterday was when to-morrow was to-day when the day after to-morrow was yesterday. On what day was the chicken hatched out?

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“It can now be revealed that the chicken referred to in the last Eureka was hatched out on a Friday. (Clearly this piece of information cannot now be of any interest to anyone.)” A fuller solution is here. (Thanks, Bruce.)

January 10, 2021February 24, 2021 | Puzzles

“Self-Explanatory”

This impressive puzzle appeared recently in the Magpie, a magazine for advanced UK-style cryptic puzzles. See the file for instructions.

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The composer explains the puzzle’s creation here. (Thanks, Gareth.)

January 9, 2021 | Puzzles

Correspondence

A problem from the Sixth International Mathematical Olympiad, 1964: Seventeen people correspond by mail, each with all the rest. They discuss only three topics, and each pair of correspondents addresses only one of these. Prove that there are at least three people who write to each other about the same topic.

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Choose one person and call her A. She corresponds with 16 other people, and since there are only three topics, she must correspond with at least six people about some one of these topics. Call that Topic 1. Now if any of her correspondents on that topic writes to another about the same subject, then we’ve found three people who are corresponding about the same topic. So assume that A’s six correspondents write to one another only about Topics 2 and 3. But now consider one of these six people; call her B. She is corresponding with the other five and addressing only two topics with them. That means that she must be addressing some particular topic with at least three of the five. Call that Topic 2. Now consider those three people. If one of the three writes to another about Topic 2, then we’ve found three people writing to each other about the same topic. And if none of the three writes to another about Topic 2, then all three of them must write to one another about Topic 3, and that proves the assertion.

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December 30, 2020December 30, 2020 | Puzzles

Black and White

gleaner retractor

A curious chess problem from the Jamaica Gleaner. White has just moved. If he retracts this move, he can play another that forces Black to checkmate White immediately. What move does he retract?

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He must just have captured a black queen on h8 with a pawn on g7, promoting that pawn to a white queen. If he now retracts that move, he can play Rh1+, which forces Black to capture the rook with the restored queen, giving mate. From Arthur Ford Mackenzie, Chess: Its Poetry and Its Prose, 1887.

December 29, 2020December 28, 2020 | Puzzles

Podcast Episode 325: Lateral Thinking Puzzles

Here are eight new lateral thinking puzzles — play along with us as we try to untangle some perplexing situations using yes-or-no questions.

See full show notes …

December 28, 2020January 4, 2021 | Podcast · Puzzles

Two Christmas Quizzes

King William’s College, on the Isle of Man, has posted the 2020 edition of “The World’s Most Difficult Quiz,” with its customary epigraph, Scire ubi aliquid invenire possis ea demum maxima pars eruditionis est (“The greatest part of knowledge is knowing where to find something”). Answers will be posted on January 20; as usual, MetaFilter is maintaining a Google spreadsheet of communal guesses.

And the Royal Statistical Society has posted its own 2020 Christmas quiz, which it describes as “brain-melting.” “You’ll need a combination of general knowledge, logic, and lateral thinking skills to successfully crack these puzzles — but as always, no specialist mathematical knowledge is required.” Solutions are due by the end of January; the top prize is £150 in Wiley book vouchers.

December 25, 2020December 24, 2020 | Puzzles

A Last Goodbye

William Friedman, the father of modern American cryptology, was fond of a cipher devised by Francis Bacon — a scheme so flexible that it could hide a message in a drawing, a piece of sheet music, almost any imaginable setting (“anything can be made to signify anything”).

In 1918 he used it to hide a message in the graduation photograph of the codebreakers’ class that he’d taught with his wife Elizebeth (click to enlarge). Some students are looking at the camera, others to the side — they’re encoding the message KNOWLEDGE IS POWER, a quotation of Bacon’s that was a favorite of Friedman’s.

Friedman kept the photo on his desk for the rest of his career, and when he died in 1969 Elizabeth had the phrase engraved on his tombstone, in a design of her own devising:

friedman inscription

In 2017, cryptographer Elonka Dunin noticed that the inscription is composed of both serif and sans-serif letters. It turns out that even this is a cipher — Elizabeth had used it to conceal the letters WFF, William’s initials. Dunin calls it “a fitting tribute, in the life of a couple who had been so dedicated to the field of codes and ciphers.”

There’s more on the Friedmans’ legacy in this NSA publication.

December 12, 2020 | Puzzles