Late Progress

Back in 2017 I wrote about the Feynman ciphers, three coded messages that had been presented as challenges to Richard Feynman in the 1950s.

Feynman couldn’t crack them, and even at the time of my post only the first of the three had been decoded — it turned out to be a transposition of the opening of Chaucer’s Canterbury Tales in Middle English. But in May 2023 David Vierra solved the other two — one turns out to be an excerpt from A.E. Housman’s 1896 poem “Terence, This is Stupid Stuff,” and the other is the start of Feynman’s 1953 paper “Atomic Theory of the λ Transition in Helium,” from The Physical Review.

Who was the “fellow scientist” who had devised these challenges? Nick Pelling thinks the most likely candidate is Paul Olum, who had been Feynman’s officemate at Los Alamos in the 1940s, but hard evidence is lacking. More at Cipher Mysteries.

(Thanks to reader Peter Dawyndt for the tip.)

March 25, 2025March 24, 2025 | Puzzles

Black and White

henle retrograde puzzle

Smith College mathematician Jim Henle published this retrograde analysis puzzle in the Mathematical Intelligencer in 2018 as part of an appreciation of Raymond Smullyan. “After many moves, the chessboard appears as above. What were the last two moves?”

	Select Click for Answer
Henle’s solution: The black king is in check, and the check must have been discovered when the white king moved from f3 to e2. But how is this possible? The white king would itself have been in check in two different ways on f3. How could Black have inflicted two checks at once? The answer is that a black pawn must have stood on f2, blocking the rook’s check. This pawn then captured some white piece on e1, promoting to a knight and inflicting a double check. So, from this position: 1. … fxe1=N+ 2. Ke2+. (Jim Henle, “The Entertainer,” Mathematical Intelligencer 40:2 [June 2018], 76-80.)

Click for Answer

March 24, 2025March 23, 2025 | Puzzles

The Polish Acquaintanceship Problem

A Polish problem longlisted for the 19th International Mathematical Olympiad in Yugoslavia, July 1977:

A room contains nine men. Among every three of them there are two who are mutually acquainted. Prove that four of them are mutually acquainted.

	Select Click for Answer
If one of the men is unacquainted with four of the others, then those four know one another and we’re done. If not, then this means that each man is mutually acquainted with at least five of the others. It can’t be exactly five in every case, because the total number of acquaintances in the room must be an even number. Hence someone is acquainted with at least six men. Among those six, three know one another, and they together with the man himself make four.

Click for Answer

March 21, 2025March 20, 2025 | Puzzles

The Liars

A problem by British puzzlist Hubert Phillips:

In writing home about an examination, five schoolgirls each made one true statement and one untrue one. The relevant passages:

Betty: Kitty was second in the examination. I was only third.
Ethel: You’ll be glad to hear that I was top. Joan was second.
Joan: I was third, and poor old Ethel was bottom.
Kitty: I came out second. Mary was only fourth.
Mary: I was fourth. Top place was taken by Betty.

In what order did they place?

	Select Click for Answer
Kitty Joan Betty Mary Ethel It’s possible to begin with any of the girls’ statements, but here’s one solution: Two girls say that Mary placed fourth. If this is not true then Kitty was second (per Kitty’s statement) and Betty first (per Mary’s statement). But if Betty is first then Joan is second (per Ethel’s statement), which is a contradiction. That means it’s true that Mary placed fourth, and it follows that Kitty is not second (per Kitty’s statement). Hence Betty placed third (per Betty’s statement), Ethel placed fifth (per Joan’s statement), and Joan placed second (per Ethel’s statement). That leaves Kitty, who by process of elimination must have placed first. From Problem Omnibus, 1960.

Click for Answer

March 18, 2025March 17, 2025 | Puzzles

Well?

A puzzle from the November-December 1994 issue of Quantum:

“If the brainteaser you solved before you solved this one was harder than the brainteaser you solved after you solved the brainteaser that you solved before you solved this one, was the brainteaser you solved before you solved this one harder than this one?”

	Select Click for Answer
Yes. (By N. Rozov.)

March 15, 2025 | Puzzles

One Way

https://commons.wikimedia.org/wiki/File:Maze_Type_Arrow.png — Image: Wikimedia Commons

A maze by Eric Fisk. Following the arrows, find a path that leads away from the star and back to it.

	Select Click for Answer
Image: Wikimedia Commons

March 12, 2025 | Puzzles

A Number Maze

https://commons.wikimedia.org/wiki/File:Maze_Type_Number.png — Image: Wikimedia Commons

By Wikimedia user Efbrazil. Begin at the star. The number at your current position tells you the number of blocks that your next jump must span. All jumps must be orthogonal. So, for example, your first jump must take you to the 1 in the lower left corner or the 2 in the upper right. What sequence of jumps will return you to the star?

	Select Click for Answer
Image: Wikimedia Commons 02/21/2025 UPDATE: There are at least three additional solutions. Label the columns A-D and the rows 1-4, like a miniature chessboard. Now: A4-A1-B1-D1-D4-D2-C2-C4-C1-C3-A3-A4 (from reader Justin Hilyard) A4–D4–B4–B2–C2–C4–C1–C3–A3–A4 (from reader Heinrich Hemme) A4-A1-A2-D2-C2-C4-C1-C3-A3-A4 (from reader Paul Wagner)

February 20, 2025February 23, 2025 | Puzzles

Wine and Roses

This portrait of Bacchus contains the images of two lovers. Where are they?

	Select Click for Answer
They’re surprisingly plain once you see them — their joined lips are just to the left of Bacchus’ nose. From “Old Puzzle Cards,” Strand, November 1899.

February 16, 2025February 15, 2025 | Puzzles

Shipshape

https://commons.wikimedia.org/wiki/File:Cmglee_Cambridge_Trinity_College_punt_names.jpg — Image: Wikimedia Commons

These are the punts of Trinity College, Cambridge, moored on the River Cam. What is the significance of their names?

	Select Click for Answer
Inspired by the college’s name, each punt evokes the number three: In the game NOUGHTS & CROSSES, players compete to place three marks in a row. A CODON is a sequence of three nucleotides in a DNA or RNA molecule. In Greek mythology, beauty was deified in the three GRACEs Euphrosyne, Aglaia, and Thalia. A HAIKU has three lines. Three WISE MONKEYs illustrate the maxim “See no evil, hear no evil, speak no evil.” NENYA is one of the three Rings of Power in J.R.R. Tolkien’s fantasy legendarium. FLUFFY is a three-headed dog in J.K. Rowling’s Harry Potter universe. Traditionally after two failures one hopes for a third LUCKY TIME. CHARLES III is king of the United Kingdom. Not shown: LITHIUM has atomic number 3. A BARYON is a subatomic particle made up of three quarks. 02/11/2025 UPDATE: Wow, yet more! The additional names include LITTLE KITTEN, CHEER, BRONZE, WISE MAN, PEACE SWEET, GRUFF, MAD GEORGE, DIABOLUS IN MUSICA, HARRY LIME, and POINT TURN. (Thanks, Dmitry.)

Click for Answer

February 11, 2025February 11, 2025 | Puzzles · Trivia

Freight

A problem from Cambridge mathematician J.E. Littlewood’s Miscellany (1953):

Is it possible to pack a cube with a finite number of smaller cubes, no two of which are the same size?

	Select Click for Answer
No. Think of the large cube as a square well with a flat bottom. If the task is possible, then this floor is tiled perfectly with cubes of different sizes. Consider the smallest of these cubes. This can’t adjoin an edge of the floor, because in that position it would be impossible to surround it with cubes larger than itself. So it lies somewhere on the expanse of the floor, surrounded by larger cubes. But now we’re back where we started. The top of this small cube forms the floor of a square well, and all the reasoning above applies again. And so on forever: At each stage we’ll be forced to supply smaller cubes, and our task will never be finished.

Click for Answer

February 3, 2025February 2, 2025 | Puzzles