Penetration

The coastline of Nova Scotia was once frequented by pirates, and people occasionally dig for buried pirate treasure. On a local radio program a few years ago I heard an interview with someone who had done a study of attempts to find pirate treasure. He claimed that in most of the cases in which treasure was actually found, it was in a place where treasure-hunters had dug before, rather than in a brand new, previously undug, location. Past diggers simply hadn’t dug deep enough. The previous digger had, in fact, often stopped just short of the treasure. If the previous digger had dug a little deeper than he did, he would have found it.

The interviewer asked him what advice he would give to treasure hunters on the basis of this study; and, producing an interesting application of induction, he lamely suggested that diggers should dig a little deeper than they in fact do. Can you see why this advice is impossible to follow?

— Robert M. Martin, There Are Two Errors in the the Title of This Book, 2002

March 12, 2026March 11, 2026 | Science & Math

Paperwork

Meek young men grow up in libraries, believing it their duty to accept the views, which Cicero, which Locke, which Bacon, have given, forgetful that Cicero, Locke, and Bacon were only young men in libraries when they wrote these books.

— Emerson, “The American Scholar,” 1838

(Daniel Dennett: “A scholar is just a library’s way of making another library.”)

Charlie’s Birthday

A puzzle by National Security Agency mathematician Stephen C., from the agency’s July 2015 Puzzle Periodical:

Charlie presents a list of 14 possible dates for his birthday to Albert, Bernard, and Cheryl.

Apr 14, 1999
Feb 19, 2000
Mar 14, 2000
Mar 15, 2000
Apr 16, 2000
Apr 15, 2000
Feb 15, 2001
Mar 15, 2001
Apr 14, 2001
Apr 16, 2001
May 14, 2001
May 16, 2001
May 17, 2001
Feb 17, 2002

He then announces that he is going to tell Albert the month, Bernard the day, and Cheryl the year.

After he tells them, Albert says, “I don’t know Charlie’s birthday, but neither does Bernard.”

Bernard then says, “That is true, but Cheryl also does not know Charlie’s birthday.”

Cheryl says, “Yes, and Albert still has not figured out Charlie’s birthday.”

Bernard then replies, “Well, now I know his birthday.”

At this point, Albert says, “Yes, we all know it now.”

What is Charlie’s birthday?

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When Albert claims that Bernard does not know Charlie’s birthday, he is saying that he knows that the correct day occurs more than once in the list. In other words, he is saying that Charlie’s birthday is not Feb 19, 2000, and the only way he could know that is that he knows that the month is not February. So by making this claim Albert has reduced the list for everyone to: Apr 14, 1999 Mar 14, 2000 Mar 15, 2000 Apr 16, 2000 Apr 15, 2000 Mar 15, 2001 Apr 14, 2001 Apr 16, 2001 May 14, 2001 May 16, 2001 May 17, 2001 When Bernard says that it is true that he does not know Charlie’s birthday, it tells everyone that even with the restricted list the correct day occurs more than once in the list. So everyone can thus eliminate May 17, 2001. Furthermore, the claim that Cheryl also does not know Charlie’s birthday is a claim that Bernard knows that the year occurs more than once on the remaining list. This rules out Apr 14, 1999, and since Bernard could only rule this out by knowing that the day was not 14, everyone can further reduce the list to: Mar 15, 2000 Apr 16, 2000 Apr 15, 2000 Mar 15, 2001 Apr 16, 2001 May 16, 2001 When Cheryl says that Albert still has not figured out Charlie’s birthday, she is telling everyone that given the new list the month occurs more than once and thus rules out May 16, 2001. This tells everyone that Cheryl knows that the year is not 2001 and the list can be reduced to: Mar 15, 2000 Apr 16, 2000 Apr 15, 2000 When Bernard then claims to know the date he is saying that the day occurs only once among the 3 remaining choices, thus telling everyone that Charlie’s birthday is Apr 16, 2000.

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March 11, 2026March 10, 2026 | Puzzles

Prospects

“The telescope makes the world smaller; it is only the microscope that makes it larger.” — G.K. Chesterton

March 10, 2026March 9, 2026 | Science & Math

“An Egg Sent Through the Post”

I send you a photograph of the empty shell of an ostrich’s egg, with the necessary Customs declaration attached by means of a string tied to a match, and inserted in one of the holes. The shell bears the addresses of the sender and receiver written in ink, and also has the postage-stamps affixed. The novelty lies in the fact that it came by the ordinary post from Port Elizabeth (S. Africa) to Whitstable, nearly seven thousand miles, exactly as seen in the photo — that is to say, with no packing whatever — and arrived in a perfectly undamaged condition.

— W.H. Reeves, in the Strand, November 1903

March 10, 2026March 9, 2026 | Oddities

The Pulfrich Effect

https://commons.wikimedia.org/wiki/File:Pulfrich_effect_pendulum.svg — Image: Wikimedia Commons

When you view a pendulum swinging laterally before your eyes, your brain understands correctly that the bob is moving in a straight line perpendicular to your line of sight. But if you put a dark filter over one eye, the bob seems to move in an ellipse, swinging somewhat closer to the screened eye.

Apparently the visual system responds more quickly to bright objects than to dim ones, so when the clear eye correctly sees the bob’s position at A, B, and C, the obscured eye sees it at A’, B’, and C’, and the brain reconciles these reports by supposing it’s at A*, B*, and C*. German physicist Carl Pulfrich first described the effect in 1922.

March 9, 2026March 8, 2026 | Oddities · Science & Math

The Devil’s Golf Course

Death Valley contains an enormous jagged salt flat produced by the evaporation of an ancient lake.

It takes its name from a 1934 National Park Service guidebook, which declares that “only the devil could play golf on such rough links.”

March 9, 2026March 8, 2026 | Oddities

Neck Deep

https://commons.wikimedia.org/wiki/File:Tie_diagram_inside-out_start.svg — Image: Wikimedia Commons

In 1999, while serving as research fellows at Cambridge University’s Cavendish Laboratory, physicists Thomas Fink and Yong Mao made a mathematical study of necktie knots. They published a summary in Nature that year and a detailed exposition in Physica A in 2000.

They found that, if knots are modeled as persistent random walks on a triangular lattice, there are exactly 85 ways to tie a tie. Of the 10 knots they scored as most aesthetic (for symmetry and balance), only four (four-in-hand, Pratt knot, half-Windsor, Windsor) are well known to Western men; interestingly, the simplest of the remainder, the unassuming small knot, above, is popular in the communist youth organization in China.

Here’s a list of the most aesthetic knots in their list.

March 8, 2026March 7, 2026 | Science & Math

Sliding Dominoes

https://commons.wikimedia.org/wiki/File:100_grid.svg — Image: Wikimedia Commons

The squares of a 9×9 board are colored as shown, and then its surface is covered with 40 dominoes. Each domino covers two orthogonally adjacent squares, and the uncovered square is a black square on the boundary.

A move shifts a domino along its length by one square, so that it covers one empty square and exposes another. Prove that, for each of the black squares on the board, there’s a sequence of moves that will uncover it.

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This solution is by Li Zhou of Polk Community College in Winter Haven, Florida. Draw an arrow on each domino that’s covering a black square, starting on the black square and continuing through the white square. Now every black square on the board (except for the one that was exposed at the start) is covered by the tail of an arrow, and each arrow points toward a neighboring black square. This provides an answer. For any black square that’s covered at the start, the arrows now indicate a route leading to the uncovered black square. Shifting the dominoes along this route will uncover the chosen black square. Will every such route find its way to the uncovered square? Yes. The alternative would be some “loop” of arrows within the diagram. Any such loop would enclose an odd number of squares, and such an interior could not be tiled by dominoes. So we know it doesn’t exist. From the 1997 Russian Olympiad for high school students, via American Mathematical Monthly, April 2004 (Problem 10960). A solution using a graph model appears in the 1999 pamphlet Mathematical Olympiads 1997-1998: Problems and Solutions From Around the World, by T. Andreescu and K. Kedlaya.

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