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?

	Select Click for Answer
This can be done intuitively. Imagine handing out the apples as described in the first case and then taking back one apple from each of the boys who received three. At that point we’ll be holding eight apples. So there are 9 boys and 26 apples.

August 13, 2021August 13, 2021 | Puzzles

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.

You can listen using the player above, download this episode directly, or subscribe on Google Podcasts, on Apple Podcasts, or via the RSS feed at https://futilitycloset.libsyn.com/rss.

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!

August 9, 2021August 8, 2021 | Podcast · Puzzles

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?

	Select Click for Answer
It’s a transposition cipher. Lay out the 336 letters into 4 rows of 84 characters: TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGCHANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASL EGASTHNGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEEFALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEA NRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODDOTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS FHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZNRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ Cut that in two and lay the two halves alongside one another: TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGC EGASTHNGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEE NRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODD FHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZ HANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASL FALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEA OTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS NRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ Interleave the rows: TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGC HANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASL EGASTHNGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEE FALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEA NRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODD OTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS FHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZ NRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ Combine the halves again and read first the odd and then the even-numbered columns: TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGCNRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODD HANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASLOTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS EGASTNHGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEEFHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZ FALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEANRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ “THE FINAL LOT OF THE LAST THREE THOUSAND WAS SENT OFF ON THURSDAY AND GOT THROUGH SAFE STOP NOTHING SEEN OF NIPASGAL STOP THINK OF THROWING THEM WELL OFF THE SCENT BY USING ANOTHER PORT NEXT TIME STOP ADVISE AGAINST LANDING CHEMICALS TILL FURTHER NOTICE STOP GREY CLOUD IN OFFING STOP WIL GIVE YOU ALL CLEAR THRICE COMMA AT NINE COMMA NINE THIRTY COMMA AND TEN THIRTY COMMA ON NIGHT BEFORE I EXPECT YOU ENDS ZZ” The method of solution is laid out on pages 148-161 of the novel. (Via Craig Bauer’s 2016 book Secret History: The Story of Cryptology.) 08/21/2021 UPDATE: Connington’s text contains six errors, which are carried through in Bauer’s exercise. I’d caught two of these in writing this post but overlooked the other four. Reader Peter Dawyndt kindly sent the ciphertext as it should have appeared: TEIIL LFILH TCETU FDHSO OENPR YYUGO HNGOF LOVTU GCHAN NOATN AEHAT ISUWE ETFST GSCAD OFRGH PELPE HASLE GASTN HGSMR LHLAR ARNIF THRDL NITFO SSWSG NYILE EFALT ODECT IESOL NTSNT COOUE AODNT IUTAI TIOON LEANR IIGOT AHNOM RINHE YLMFD STTTS MANHH OFEII ETODD OTPCA MOTIE FMONG IMCLA TTCHB YIMNN ETROX EMCOU VSFHA ELMPN NCTAW ETRWO OAHEE IYCNA OIRBT RTXET PEIZN RSCSA TIKOH NITHT EMFNE NNRUO GOTGP ENETP SYANS Z Also, I introduced one typo in the plaintext — WIL should be WILL. Sorry for these errors; this cipher must have frustrated Connington’s readers. (Thanks, Peter.)

Click for Answer

It’s a transposition cipher. Lay out the 336 letters into 4 rows of 84 characters:

TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGCHANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASL
EGASTHNGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEEFALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEA
NRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODDOTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS
FHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZNRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ

Cut that in two and lay the two halves alongside one another:

TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGC
EGASTHNGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEE
NRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODD
FHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZ

HANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASL
FALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEA
OTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS
NRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ

Interleave the rows:

TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGC
HANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASL
EGASTHNGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEE
FALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEA

NRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODD
OTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS
FHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZ
NRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ

Combine the halves again and read first the odd and then the even-numbered columns:

TEIILLFILHTCETUFDHSOOENPRYYUGOHNGOFLOVTUGCNRIIGOTAHNOMRINHEYLMFDSTTTSMANHHOFEIIETODD
HANNOATNAEHATISUWEETFSTGSCADOFRGHPELPEHASLOTPCAMOTIEFMONGIMCLATTCHBYIMNNETROXEMCOUVS
EGASTNHGSMRLHLARARNIFTHRDLNITFOSSWSGNYILEEFHAELMPNNCTAWETRWOOAHEEIYCNAOIRBTRTXETPEIZ
FALTODECTIESOLNTSNTCOOUEAODNTIUTAITIOONLEANRSCSATIKOHNITHTEMFNENNRUOGOTGPENETPSYANSZ

“THE FINAL LOT OF THE LAST THREE THOUSAND WAS SENT OFF ON THURSDAY AND GOT THROUGH SAFE STOP NOTHING SEEN OF NIPASGAL STOP THINK OF THROWING THEM WELL OFF THE SCENT BY USING ANOTHER PORT NEXT TIME STOP ADVISE AGAINST LANDING CHEMICALS TILL FURTHER NOTICE STOP GREY CLOUD IN OFFING STOP WIL GIVE YOU ALL CLEAR THRICE COMMA AT NINE COMMA NINE THIRTY COMMA AND TEN THIRTY COMMA ON NIGHT BEFORE I EXPECT YOU ENDS ZZ”

The method of solution is laid out on pages 148-161 of the novel.

(Via Craig Bauer’s 2016 book Secret History: The Story of Cryptology.)

08/21/2021 UPDATE: Connington’s text contains six errors, which are carried through in Bauer’s exercise. I’d caught two of these in writing this post but overlooked the other four. Reader Peter Dawyndt kindly sent the ciphertext as it should have appeared:

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

Also, I introduced one typo in the plaintext — WIL should be WILL. Sorry for these errors; this cipher must have frustrated Connington’s readers. (Thanks, Peter.)

August 8, 2021August 21, 2021 | Puzzles

Black and White

baird chess problem

By Edith Baird. White to mate in two moves.

	Select Click for Answer
1. Bf4! 1. … exf4 2. Qe8# 1. … dxe3 2. Qa4# 1. … f2 2. Kxc7# 1. … d2 2. f6# From Field, July 16, 1898.

August 6, 2021August 5, 2021 | Puzzles

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?

	Select Click for Answer
Imagine the field as a checkerboard with white corners. Shoot the 112 black squares, then the 113 white squares, then the black squares again. Two of these shots must hit the tank; in the worst case this will take 337 shots. To see that this is best, imagine covering the field with 112 dominoes, leaving one square uncovered, and suppose that the tank is restricted to its domino. We can study each domino independently: Each will require three shots, and the uncovered square requires a shot as well. So the lower bound is 3 × 112 + 1 = 337.

Click for Answer

August 4, 2021August 3, 2021 | Puzzles

The Wine-Bin

A puzzle by Claude Gaspar Bachet de Méziriac, from 1612, via Henry Dudeney:

A gentleman had a wine-bin of eight compartments, as in the illustration, containing 60 bottles, arranged as shown. His dishonest servant stole 4 bottles and rearranged the remainder. The gentleman noticed that the bottles had been redistributed, but as there were still 21 bottles on every side he innocently concluded that all the 60 were there. The servant, emboldened by his success, again stole 4 bottles and rearranged the remainder without discovery. In fact, on two more occasions he repeated his theft of 4 bottles, always leaving the remainder so arranged symmetrically that there were 21 on every side. How did he arrange them on the four occasions so as to steal the 16 bottles?

	Select Click for Answer
(Henry E. Dudeney, “The World’s Best Puzzles,” Strand 36:216 [January 1909], 692-700; Henry E. Dudeney, “The World’s Best Puzzles — Solutions,” Strand 36:217 [February 1909], 113-116.)

August 2, 2021August 1, 2021 | Puzzles

Implementia

A puzzle by Yoshinao Katagiri: A boy and a girl played rock paper scissors 10 times. Altogether the boy played rock three times, scissors six times, and paper once, and the girl played rock twice, scissors four times, and paper four times (though, in each case, the order of these plays is unknown). There were no ties. Who won?

	Select Click for Answer
Since there were no ties, each time the boy played scissors the girl must have played rock or paper. The girl’s record shows us that he lost two of those six rounds and won four. Likewise, in the four rounds in which the girl played scissors, the boy played rock three times and paper once. She lost three times and won once. Altogether the boy won seven rounds and lost three. (Via Nobuyuki Yoshigahara’s Puzzles 101: A Puzzlemaster’s Challenge, 2004.)

Click for Answer

July 28, 2021July 27, 2021 | Puzzles

Fixing a Point

A problem proposed by Richard Hoshino and Sarah McCurdy for Crux Mathematicorum, September 2008:

Five points lie on a line. Here are the 10 distances between pairs of points, listed from smallest to largest:

2, 4, 5, 7, 8, k, 13, 15, 17, 19

What’s k?

	Select Click for Answer
This solution is by Gustavo Krimker of Universidad CAECE, Buenos Aires. The distance between the first and fifth points is 19. That means that 19 is also the sum of each of three pairs of intermediate distances: (the distance from the first point to the second point) + (the distance from the second point to the fifth point) = 19 (the distance from the first point to the third point) + (the distance from the third point to the fifth point) = 19 (the distance from the first point to the fourth point) + (the distance from the fourth point to the fifth point) = 19 Looking over the list of distances we’ve been given, the only pairs that add to 19 are (2, 17) and (4, 15). So those pairs correspond to two of the bulleted sums above. What’s the third pair? It must be (7, k) or (8, k). (It can’t be (5, k) or (k, 13) because we know that k falls between 8 and 13 and so those pairs can’t be made to total 19.) And if the third case is (7, k) or (8, k), then k must be 11 or 12. Now, 17 is the second-greatest distance on the list; that means it’s the distance between one of the end points in the row of five and the penultimate point on the opposite side. Either way, that means that 17 is the sum of two pairs of intermediate distances, using the same reasoning as above. One of those pairs is (4, 13), and the other must be (5, k), (7, k), or (8, k). That means that k must be 12, 10, or 9. The only one of these that matches a candidate above is 12. (Ian VanderBurgh, “Mathematical Mayhem,” Crux Mathematicorum 34:5 [September 2008], 275-286.)

Click for Answer

July 20, 2021July 19, 2021 | Puzzles

Black and White

carney chess problem

By E. Carney Jr. White to mate in two moves.

	Select Click for Answer
1. Rb3! and the queen will mate. From the Southern Trade Gazette, via Charles F. Wadworth’s Unique Chess Problems, 1890. 07/20/2021 UPDATE: This is cooked. 1. Qd6+ works just as well and is simpler. (Thanks, Travis.) 07/20/2021 UPDATE: Blimey, doubly cooked! 1. Rc3 works too. (Thanks, Chris.) And the pawn seems to have no role in any of this, not even in Carney’s intended solution. Not sure how Wadworth can have chosen this without noticing the errors — but I guess I just did the same thing!

Click for Answer

July 19, 2021July 20, 2021 | Puzzles

Enlightenment

You’re in a dark room. The only light comes from an old LED digital alarm clock with four seven-segment displays. The time is displayed in 24-hour format, HH:MM (no seconds), and the leading digit is blank if not used. How much time passes between the room’s darkest state and its lightest?

	Select Click for Answer
18 hours 57 minutes (1:11 to 20:08). (Thanks, Nick.) 07/16/2021 UPDATE: Properly speaking the interval is one minute less than that, because it extends from the last moment of 1:11 to the first of 20:08. Also, given my wording, the answer might equally be 5 hours 3 minutes, as that’s the interval “between” 20:08 and 1:11. (Thanks, Michael and Joe.)

July 15, 2021July 16, 2021 | Puzzles