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A puzzle proposed by David L. Silverman in the Fall 1963 issue of Pi Mu Epsilon Journal:

The points of the plane are divided into two sets. Prove that at least one set contains the vertices of a rectangle.

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This solution is by John E. Ferguson of Oregon State University: Any straight line must contain at least four points P1, P2, P3, P4 that fall into one of the sets (this must be true of one set or the other). Call that set S1. Now consider the configuration above (the distances between the points are out of scale). If S1 doesn’t contain the vertices of a rectangle, then at least three points in the A group and at least three points in the B group must belong to the other set, S2. And in that case there is an S2 rectangle. (“Problem Department,” Pi Mu Epsilon Journal 3:9 [Fall 1963], 471-476.)

November 27, 2020November 26, 2020 | Puzzles

Shikaku

Japanese publisher Nikoli produces this geometric logic puzzle. Can you divide this grid into rectangular and square pieces such that each piece contains exactly one number and each number reflects its piece’s area?

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Image: Wikimedia Commons

November 24, 2020November 23, 2020 | Puzzles

Black and White

baird chess problem

By Mrs. W.J. Baird, from the Norwich Mercury. White to mate in two moves.

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The name of the newspaper is a clue. Convert the N to an M by moving 1. Re6 and White will mate next move. Via S.S. Blackburne, Terms and Themes of Chess Problems, 1907.

November 22, 2020November 21, 2020 | Puzzles

Another First

Jules Verne’s 1882 novel La Jangada tells the story of Joam Dacosta, a Brazilian man wrongly accused of theft and murder. In Book Two his friends struggle to save him by solving a cryptogram, whose last paragraph is given in the text:

Phyjslyddqfdzxgasgzzqqehxgkfndrxujugiocytdxvksbxhhuypo
hdvyrymhuhpuydkjoxphetozsletnpmvffovpdpajxhyynojyggayme
qynfuqlnmvlyfgsuzmqiztlbqgyugsqeubvnrcredgruzblrmxyuhqhp
zdrrgcrohepqxufivvrplphonthvddqfhqsntzhhhnfepmqkyuuexktog
zgkyuumfvijdqdpzjqsykrplxhxqrymvklohhhotozvdksppsuvjhd.

In the end this works out to:

Le véritable auteur du vol des diamants et de l’assassinat des soldats qui escortaient le convoi, commis dans la nuit du vingt-deux janvier mil huit cent vingt-six, n’est donc pas Joam Dacosta, injustement condamné à mort; c’est moi, le misérable employé de l’administration du district diamantin; oui, moi seul, qui signe de mon vrai nom, Ortega.

In the article linked below, Miami University mathematician Frederick Gass explains rigorously how the cipher might be solved. In the novel, Judge Jarriquez has a brainstorm: He learns that the writer might have been named Ortega, guesses that the declaration might end with that signature, and works out the rest from there.

“By virtue of this solution, Jules Verne is credited with the first published exposition of the probable word method for Gronsfeld ciphers.”

(Frederick Gass, “Solving a Jules Verne Cryptogram,” Mathematics Magazine 59:1 [February 1986], 3-11.)

November 18, 2020November 17, 2020 | Literature · Puzzles

The Hard Way

https://youtu.be/Y2jiQXI6nrE?t=1010s

This is great — Eugene Wigner tells the story of Max Born giving the “two bikes and a fly” puzzle to John von Neumann (it starts at 16:50).

(Via Tamás Görbe, from an old VHS video digitized by Robert Klips.)

November 17, 2020November 16, 2020 | Puzzles · Science & Math

A Captivating Entertainment

Visitors to La Granja, Philip V’s retreat overlooking Madrid, may have found this unusual labyrinth a bit too involving. It’s a “vortex maze”: It “leads you directly to the centre, but its spiralling options for exit are your undoing,” explain Angus Hyland and Kendra Wilson in The Maze: A Labyrinthine Compendium (2018). “On paper, it is a beautiful design, and in reality it is almost twice as large as the ‘record-breaking’ maze at Longleat, made 250 years later.”

Philip’s French landscapers drew it from La théorie et la pratique du jardinage, the best-selling gardening manual published by Antoine Joseph Dezallier d’Argenville in 1709.

November 14, 2020 | History · Puzzles

The Church Cipher

In 1775, an enciphered message was intercepted en route from colonial physician Benjamin Church to a British officer in Boston. Part of the message appears below. What does it say?

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It’s a simple substitution cipher: So this sentence reads, “Should Britain declare War against the Colonies, they are lost forever.” A court of inquiry found that Church had carried on a criminal correspondence with the enemy. He was jailed until 1778, when he sailed from Boston and was never heard from again. (From the National Park Service.)

November 5, 2020November 4, 2020 | History · Puzzles

Black and White

penrose chess problem

Physicist Roger Penrose devised this problem in 2017 to illustrate the difference between artificial and human intelligence. James Tagg, leader of the Penrose Institute, told the Telegraph, “A human looking at it for a short while will ‘see’ what white must, and more particularly, must not do, and use very little energy to decide this. But, for a computer, the puzzle requires an enormous number of calculations, far too many for even today’s supercomputers.” What is White’s insight?

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Penrose wrote, “If you put this puzzle into a chess computer it just assumes a black win because of the number of pieces and positions, but a human will look at this and know quickly that is not the case.” This is because most of Black’s pieces are entombed on the queenside; his bishops are free to move, but they can neither free the other black pieces nor checkmate the white king. Penrose explained: If both players understand the position and play perfectly there is no way either player can win – black can always keep its bishops together so the white king cannot pick them off, and the 50-move rule would result in a draw. Human players understand this and don’t need to calculate the position. However, we can see that Fritz (one of the better AI chess programs) looks ahead 20 moves and thinks black is winning. Fritz does not understand the nature of the position: It sees the material advantage and thinks it is ahead. The question is whether the human playing as white can use this lack of understanding to win. A small number of you ran the puzzle on chess programs and experimented with this. Although we maintain that chess programs (such as Fritz and Stockfish) do not ‘understand’ the position, they played correctly and achieved a draw despite believing they had a likely win. However, some programs set to a low level of skill were so convinced they should win, they refused the draw through repeated position. They accepted losing the bishops one after another until they blundered into a losing position, allowing the white pawn on the sixth rank to be advanced and turned into a queen. To make this actually happen requires a bit of trickery and a great deal of patience. “We know that there are things that the human mind achieves that even the most powerful supercomputer cannot but we don’t know why,” he added. “If we find out how humans differ from computers then it could have profound sociological implications. People get very depressed when they think of a future where robots or computers will take their jobs, but it might be that there are areas where computers will never be better than us, such as creativity.”

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Penrose wrote, “If you put this puzzle into a chess computer it just assumes a black win because of the number of pieces and positions, but a human will look at this and know quickly that is not the case.” This is because most of Black’s pieces are entombed on the queenside; his bishops are free to move, but they can neither free the other black pieces nor checkmate the white king. Penrose explained:

If both players understand the position and play perfectly there is no way either player can win – black can always keep its bishops together so the white king cannot pick them off, and the 50-move rule would result in a draw. Human players understand this and don’t need to calculate the position.

However, we can see that Fritz (one of the better AI chess programs) looks ahead 20 moves and thinks black is winning. Fritz does not understand the nature of the position: It sees the material advantage and thinks it is ahead. The question is whether the human playing as white can use this lack of understanding to win. A small number of you ran the puzzle on chess programs and experimented with this.

Although we maintain that chess programs (such as Fritz and Stockfish) do not ‘understand’ the position, they played correctly and achieved a draw despite believing they had a likely win. However, some programs set to a low level of skill were so convinced they should win, they refused the draw through repeated position. They accepted losing the bishops one after another until they blundered into a losing position, allowing the white pawn on the sixth rank to be advanced and turned into a queen. To make this actually happen requires a bit of trickery and a great deal of patience.

“We know that there are things that the human mind achieves that even the most powerful supercomputer cannot but we don’t know why,” he added. “If we find out how humans differ from computers then it could have profound sociological implications. People get very depressed when they think of a future where robots or computers will take their jobs, but it might be that there are areas where computers will never be better than us, such as creativity.”

November 2, 2020November 1, 2020 | Puzzles

Euphony

A charming puzzle from Crux Mathematicorum, December 2004:

If all plinks are plonks and some plunks are plinks, which of these statements must be true?

X: All plinks are plunks.
Y: Some plonks are plunks.
Z: Some plinks are not plunks.

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(Solution by Geneviève Lalonde, Massey, Ontario:) In this diagram the various regions are numbered. Since all plinks are plonks, there can be nothing in region 1 or 6. And since some plunks are plinks, the regions 6 and 7, considered together, cannot be empty. Combining these conclusions shows that region 7 can’t be empty. Now: Statement X says that regions 1 and 4 are both empty. We know nothing about region 4, so we can’t say whether this statement is true. Statement Y says that regions 5 and 7, considered together, can’t be empty. We know that region 7 is not empty, so this statement is true. Statement Z says that regions 1 and 4, considered together, can’t be empty. We’ve established that region 1 is empty, but again, we know nothing about region 4, so we don’t know whether this statement is true. Hence the only statement that must be true is Y.

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