# Exam Week

A problem submitted by the United States and shortlisted for the 16th International Mathematical Olympiad, Erfurt-Berlin, July 1974:

Alice, Betty, and Carol took the same series of examinations. There was one grade of A, one grade of B, and one grade of C for each examination, where A, B, C are different positive integers. The final test scores were

Alice: 20
Betty: 10
Carol: 9

If Betty placed first in the arithmetic examination, who placed second in the spelling examination?

# To Town

A problem from the 1949 problems drive of the Archimedeans, the mathematical society of Cambridge University:

A farmer lives in a cottage 4/17 of a mile from a main road. There is a lane leading from his farm to the nearest point Q on the road. The road is straight running north and south, and there is a village two miles south of Q at which he keeps a bicycle. He wishes to go to a town on the road four miles north of Q. He can walk across the fields surrounding the roads at 1 1/2 miles per hour, but along the roads he can walk at 3 1/2 miles per hour. He can cycle at 14 miles per hour. Should he collect his bicycle in order to get to the town from his farm as quickly as possible?

# \$50 for All

A puzzle by Ben H., a systems engineer at the National Security Agency, from the agency’s August 2016 Puzzle Periodical:

At a work picnic, Todd announces a challenge to his coworkers. Bruce and Ava are selected to play first. Todd places \$100 on a table and explains the game. Bruce and Ava will each draw a random card from a standard 52-card deck. Each will hold that card to his/her forehead for the other person to see, but neither can see his/her own card. The players may not communicate in any way. Bruce and Ava will each write down a guess for the color of his/her own card, i.e. red or black. If either one of them guesses correctly, they both win \$50. If they are both incorrect, they lose. He gives Bruce and Ava five minutes to devise a strategy beforehand by which they can guarantee that they each walk away with the \$50.

Bruce and Ava complete their game and Todd announces the second level of the game. He places \$200 on the table. He tells four of his coworkers — Emily, Charles, Doug and Fran — that they will play the same game, except this time guessing the suit of their own card, i.e. clubs, hearts, diamonds or spades. Again, Todd has the four players draw cards and place them on their foreheads so that each player can see the other three players’ cards, but not his/her own. Each player writes down a guess for the suit of his/her own card. If at least one of them guesses correctly, they each win \$50. There is no communication while the game is in progress, but they have five minutes to devise a strategy beforehand by which they can be guaranteed to walk away with \$50 each.

For each level of play — 2 players or 4 players — how can the players ensure that someone in the group always guesses correctly?

# Paint Job

Given a standard chessboard, you can choose any rank or file and repaint each of its squares to the opposite color (white squares turn black, and black squares turn white). By doing this repeatedly, is it possible to produce a board with 63 white squares and one black square?

# Black and White

A tidy problem by G.A.W. Cumming. White to mate in two moves.

# The Picnic

We’re giving out apples to a group of boys. If we distribute the entire supply, then every boy will get three, except for one, who will get two. If instead we give each boy two apples, then we’ll have eight apples left over. How many apples are there altogether?

# Podcast Episode 353: Lateral Thinking Puzzles

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

Intro:

Lili McGrath’s 1915 “floor polisher” is a pair of slippers connected by a cord.

Eighteenth-century English landowners commissioned custom ruins.

The sources for this week’s puzzles are below. In some cases we’ve included links to further information — these contain spoilers, so don’t click until you’ve listened to the episode:

Puzzle #1 is from listener Moxie LaBouche.

Puzzle #2 is from listener Cheryl Jensen, who sent this link.

Puzzle #3 is from listener Theodore Warner. Here’s a link.

Puzzle #4 is from listener David Morgan.

Puzzle #5 is from listener Bryan Ford, who sent these links.

Puzzle #6 is from listener John Rusk, who sent this link.

Please consider becoming a patron of Futility Closet — you can choose the amount you want to pledge, and we’ve set up some rewards to help thank you for your support. You can also make a one-time donation on the Support Us page of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

# Skulduggery

This secret message appears in J.J. Connington’s 1933 novel Tom Tiddler’s Island:

TEIIL LFILH TCETU FDHSO OENPR YYUGO HNGOF
LOVTU GCHAN NOATN AEHAT ISUWE ETFST GSCAD
OFRGH PELPE HASLE GASTH HGSMR LHLAR ARNIF
THRDL NITFO SSWSG NYILE EFALT ODECT IESOL
NTSNT COOUE AODNT IUTAI TIOON LEANR IIGOT
AHNOM FINHE YLMFD ATTTS MANHH OFEII ETODD
OTPCA MOTIE FMONG IMCLA TTCHB YIMNN ETROX
EMCOU VSFHE ELMPN NCTAW ETRWO OAHEE IYCNA
OIRBT RTXET PEIZN RSCSA TIKOH NITHT EMFNE
NNRUO GOTGP ENETP SYANS Z

What does it mean?

# Black and White

By Edith Baird. White to mate in two moves.

# Tank Hunt

A puzzle from Daniel J. Velleman and Stan Wagon’s excellent 2020 problem collection Bicycle or Unicycle?:

Before you is a field of 225 squares arranged in a 15×15 grid. One of the squares contains a perfectly camouflaged tank that you’re trying to destroy. You have a weapon that will destroy one square of the grid with each shot, but it takes two shots to destroy the tank, and you know that when the tank has been hit the first time (and only then) it will flee invisibly to an adjacent square (horizontally or vertically). What’s the minimum number of shots you’ll need to be sure of destroying it?