Quickie

CHAIN is to INGOT as MILLS is to … ?

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SORRY. Advance each letter six places through the alphabet. (Thanks, Nick.)

December 8, 2020December 7, 2020 | Puzzles

The Substitute

Between July 2015 and October 2018, the National Security Agency offered a monthly puzzle written by a staff member. Here’s the puzzle for October 2015, created by applied research mathematician Ben E.:

Kurt, a math professor, has to leave for a conference. At the airport, he realizes he forgot to find a substitute for the class he was teaching today! Before shutting his computer off for the flight, he sends an email: “Can one of you cover my class today? I’ll bake a pie for whomever can do it.” He sends the email to Julia, Michael, and Mary Ellen, his three closest friends in the math department, and boards the plane.

As Kurt is well-known for his delicious pies, Julia, Michael, and Mary Ellen are each eager to substitute for him. Julia, as department chair, knows which class Kurt had to teach, but she doesn’t know the time or building. Michael plays racquetball with Kurt so he knows what time Kurt teaches, but not the class or building. Mary Ellen helped Kurt secure a special projector for his class, so she knows what building Kurt’s class is in, but not the actual class or the time.

Julia, Michael, and Mary Ellen get together to figure out which class it is, and they all agree that the first person to figure out which class it is gets to teach it (and get Kurt’s pie). Unfortunately the college’s servers are down, so Julia brings a master list of all math classes taught that day. After crossing off each of their own classes, they are left with the following possibilities:

Calc 1 at 9 in North Hall
Calc 2 at noon in West Hall
Calc 1 at 3 in West Hall
Calc 1 at 10 in East Hall
Calc 2 at 10 in North Hall
Calc 1 at 10 in South Hall
Calc 1 at 10 in North Hall
Calc 2 at 11 in East Hall
Calc 3 at noon in West Hall
Calc 2 at noon in South Hall

After looking the list over, Julia says, “Does anyone know which class it is?” Michael and Mary Ellen immediately respond, “Well, you don’t.” Julia asks, “Do you?” Michael and Mary Ellen both shake their heads. Julia then smiles and says, “I do now. I hope he bakes me a chocolate peanut butter pie.”

Which class does Kurt need a substitute for?

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1) Since Julia only knows the class name, the only way she could immediately know is if it was Calc 3. Since Michael and Mary Ellen both know that Julia doesn’t know, that means they know the class isn’t Calc 3. Since Michael only knows the time, that means the class can’t be at noon. Because Mary Ellen only knows the building, that means the building can’t be West Hall. That leaves only the following possibilities: Calc 1 at 9 in North Hall Calc 1 at 10 in East Hall Calc 2 at 10 in North Hall Calc 1 at 10 in South Hall Calc 1 at 10 in North Hall Calc 2 at 11 in East Hall 2) Since Michael doesn’t know which class it is, that means the time can’t be 9 or 11. Since Mary Ellen doesn’t know either, the class can’t be in South Hall. That leaves only three possibilities: Calc 1 at 10 in East Hall Calc 2 at 10 in North Hall Calc 1 at 10 in North Hall 3) At this point, Julia now says she knows the answer. Since there are two Calc 1 classes, it must be that Julia knows the class is Calc 2. Thus, the class is Calc 2 at 10 in North Hall.

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December 6, 2020 | Puzzles

Quarters

USC mathematician Solomon Golomb offered this puzzle in his column, “Golomb’s Gambits,” in Johns Hopkins Magazine. How can you dissect this figure into four congruent pieces?

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Donald Savage found this solution in a computer search described in the July-August 1999 issue of MIT Technology Review. It appears to be unique. 12/08/2020 UPDATE Not unique! Reader Irit Rubin came up with this alternate solution: (Thanks, Irit.)

December 5, 2020December 8, 2020 | Puzzles

After You

A problem from Canada’s 2003 Hypatia contest:

Xavier and Yolanda are playing a game. They begin with two piles of three coins each and take turns; on each turn a player removes one or more coins from any one pile. The winner is the player who takes the very last coin. Xavier always goes first, but Yolanda has a strategy that ensures that she will always win. What is it?

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She emulates each of Xavier’s moves in the opposite pile. If he takes one coin from Pile 1 then she takes one coin from Pile 2, and so on. This forces him to empty one pile first, whereupon she can empty the other. (Via Robert Bilinski, “Skoliad,” Crux Mathematicorum 31:1 [January 2005], 1-12.)

December 4, 2020 | Puzzles

A Christmas Puzzle

University of Cambridge mathematician Matthew Scroggs has post his annual Christmas puzzle: Behind each day in an advent calendar lies a puzzle with a three-digit answer. Combined, the answers reveal a series of compass directions that lead to Santa’s house. Ten randomly selected people who solve all the puzzles and find Santa’s house will receive prizes. More info here.

December 1, 2020November 30, 2020 | Puzzles

The Logic House

https://commons.wikimedia.org/wiki/File:Pt-logic-house1.png — Image: Wikimedia Commons

Can you draw this figure without lifting your pencil from the paper?

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Image: Wikimedia Commons 12/24/2020 UPDATE: A number of readers pointed out that the figure can be drawn without crossing a line with the pencil: Image: Wikimedia Commons

November 29, 2020December 24, 2020 | Puzzles

Foursquare

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.)

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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