Stretch Goals

stretch goals puzzle

Two circles intersect. A line AC is drawn through one of the intersection points, B. AC can pivot around point B — what position will maximize its length?

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Call the other intersection point D and draw AD and DC. All angles inscribed in a circle and subtended by the same chord are equal, so angle BAD retains the same measure as A travels around its circle. Similarly, angle BCD remains the same as C travels around its circle. This means that triangle ADC will always have the same shape: As line AC pivots around B, triangle ADC will vary in size but remain self-similar. So which position will maximize its size? AD and DC are chords of their respective circles, and the longest chord is a diameter. So turn the triangle until both of these lines are diameters (this will happen simultaneously); at that point triangle ADC will reach its maximum size and line AC its maximum length. From Mogens Larsen in Richard Guy and Robert Woodrow, eds., The Lighter Side of Mathematics, 1994.

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October 13, 2014October 19, 2014 | Puzzles · Science & Math

Black and White

morphy chess problem

A chess problem by Paul Morphy. White to mate in two moves.

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1. Ra6! From the New York Clipper, 1856.

October 9, 2014October 19, 2014 | Puzzles

Dirty Work

A puzzle by Pierre Berloquin:

Timothy, Urban, and Vincent are digging identical holes in a field.

When Timothy and Urban work together, they dig 1 hole in 4 days.
When Timothy and Vincent work together, they dig 1 hole in 3 days.
When Urban and Vincent work together, they dig 1 hole in 2 days.

Working alone, how long does it take Timothy to dig one hole?

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In one day: Timothy and Urban dig 1/4 hole. Timothy and Vincent dig 1/3 hole. Urban and Vincent dig 1/2 hole. If we made a clone of Timothy and set all four workers digging for one day, they’d dig 1/4 + 1/3 = 7/12 hole. Since Urban and Vincent working together dig 6/12 hole per day, Timothy and his clone dig 1/12. So Timothy alone digs 1/24 hole per day, and can dig one hole in 24 days.

October 1, 2014October 12, 2014 | Puzzles

A Poetic Puzzle

On Oct. 25, 1875, Lewis Carroll sent this verse to Mrs. J. Chataway, mother of one of his child-friends, Gertrude. “They embody, as you will see, some of my recollections of pleasant days at Sandown”:

Girt with a boyish garb for boyish task,
Eager she wields her spade — yet loves as well
Rest on a friendly knee, the tale to ask
That he delights to tell.

Rude spirits of the seething outer strife,
Unmeet to read her pure and simple spright,
Deem, if you list, such hours a waste of life,
Empty of all delight!

Chat on, sweet maid, and rescue from annoy
Hearts that by wiser talk are unbeguiled!
Ah, happy he who owns that tenderest joy,
The heart-love of a child!

Away, fond thoughts, and vex my soul no more!
Work claims my wakeful nights, my busy days:
Albeit bright memories of that sunlit shore
Yet haunt my dreaming gaze!

He asked her leave to have it published. The child in the verse is not named — why should he feel obliged to ask permission?

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It’s an acrostic. “I can’t feel sure from your letter whether you have or have not noticed that the verses embody her name,” he wrote on Oct. 28. “They do it in two ways — by letters, and by syllables — the only acrostic of that kind I have ever seen. Will that make any difference in the leave you give me to print the verses? If I print them, I shan’t tell anyone it is an acrostic — but someone will be sure to find it out before long.”

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September 25, 2014October 5, 2014 | Poems · Puzzles

Black and White

healey chess problem

By Francis Healey. White to mate in two moves.

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The subtle 1. Rd8! forces 1. … Kd3 2. Nc5#. From A Collection of Two Hundred Chess Problems, 1866.

September 17, 2014September 28, 2014 | Puzzles

Seeking Stability

You’re standing in a room with an uneven floor. Before you is a square table with four legs. The table wobbles, but by turning it gradually you manage to find a position in which all four feet are supported, eliminating the wobble (though now the tabletop isn’t level).

You wonder: Is this always possible? Assuming that the four legs are of equal length and that the surface of the floor varies smoothly, is it always possible to position a four-legged table so that all four legs are supported?

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Yes, it is. Label the legs A, B, C, and D (going clockwise), and position the table so that leg A is up in the air and B, C, and D are touching the floor. Call this Position 1. Now imagine that the floor is made of sand. Leaving C and D on the surface, push B into the sand until A touches the floor. Call this Position 2. Now return to Position 1 and, instead of pushing down, rotate the table clockwise, keeping legs B, C, and D in contact with the floor. Because the legs are of equal length, we know that, as you turn the table through 90 degrees toward Position 2, leg A will at some point pass beneath the surface of the floor. So there must be some intermediate point at which all four legs just touch the floor, and the table is stable.

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September 13, 2014September 21, 2014 | Puzzles

The Conway Immobilizer

Three positions, “left,” “middle,” and “right,” are marked on a table. Three cards, an ace, a king, and a queen, lie face up in some or all three of the positions. If more than one card occupies a given position then only the top card is visible, and a hidden card is completely hidden; that is, if only two cards are visible then you don’t know which of them conceals the missing card.

Your goal is to have the cards stacked in the left position with the ace on top, the king in the middle, and the queen on the bottom. To do this you can move one card at a time from the top of one stack to the top of another stack (which may be empty).

The problem is that you have no short-term memory, so you must design an algorithm that tells you what to do based only on what is currently visible. You can’t recall what you’ve done in the past, and you can’t count moves. An observer will tell you when you’ve succeeded. Can you devise a policy that will meet the goal in a bounded number of steps, regardless of the initial position?

“It’s tricky to design an algorithm that makes progress, avoids cycling, and doesn’t do something stupid when it’s about to win,” wrote Dartmouth mathematician Peter Winkler in sharing this puzzle in his book Mathematical Puzzles: A Connoisseur’s Collection (2003). It’s called “The Conway Immobilizer” because it originated with legendary Princeton mathematician John H. Conway and because it’s said to have immobilized one solver in his chair for six hours.

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If there’s an empty slot, move a card to the right (“around the corner,” if necessary) to fill it. Exception: If the position is king-blank-ace or king-ace-blank, place the ace on the king. If all three cards are visible, move the card to the right of the queen one space to the right (again, “around the corner” if necessary). Exception: If the queen is on the left, place the king on the queen. “Amazingly, the algorithm can be generalized to work with any (fixed, known) number of cards, still in three stacks,” Winkler writes. See his book for a five-rule algorithm that will reliably sort a 52-card deck.

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September 12, 2014September 21, 2014 | Puzzles

The Amazing Sand Counter

A man presents himself as the The Amazing Sand Counter. He claims that if you put some quantity of sand into a bucket, he will know at a glance how many grains there are, but he won’t tell you the number. Can you devise a test that can verify this ability without telling you anything that you don’t already know? You can ask the Sand Counter to leave the room or turn away, for example, and you can ask him questions. How can you convince yourself that he knows how many grains of sand are in the bucket when he won’t actually tell you the number?

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This is from Dennis Elliott Shasha’s 1992 book Codes, Puzzles, and Conspiracy, via Barry Cipra’s 2004 collection Tribute to a Mathemagician. Shasha wanted to find the simplest possible zero-knowledge proof — a proof in which a Prover convinces a Verifier that the Prover knows something without ever revealing the thing itself. Shasha’s solution is this: Fill the bucket with sand, display it to the Sand Counter, then ask him to leave the room. Remove some countably small number of grains. When the Sand Counter returns, ask him how many grains have been removed from the bucket. Then count the removed grains to verify this. “The Verifier can repeat this test until he is convinced that the Sand Counter is either incredibly lucky (zero-knowledge proofs are often probabilistic in nature) or is telling the truth,” Shasha writes. This actually works both ways — it also convinces the Sand Counter that the Verifier knows the total number of grains in the bucket, even though he has never revealed it. “The Sand Counter is happy to answer these questions as long as the number of grains removed is tiny compared to the number in the bucket,” Shasha writes. “This way he can be sure that the Verifier has learned how many grains there are.”

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September 11, 2014September 6, 2024 | Puzzles

Where’s the Father?

A mother is 21 years older than her son. Six years from now, she will be five times his age. Where’s the father?

I won’t give the answer to this one — if you do the math, you’ll know precisely where he is.

September 8, 2014September 7, 2014 | Puzzles

A Painting Conundrum

painting conundrum

From Stephen Barr’s Experiments in Topology (1989) via Miodrag Petkovic’s Mathematics and Chess (1997):

This apartment contains eight rooms, each measuring 9 square meters, except for the top one, which measures 18 square meters. You have enough red paint to cover 27 square meters, enough yellow paint to cover 27 square meters, enough green paint to cover 18 square meters, and enough blue paint to cover 9 square meters. Can you paint the eight floors in four colors so that each room neighbors rooms of the other three colors?

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The task can’t be done with the colors given, but there’s a trick: If you mix 1/3 of the red paint into the blue paint you’ll produce enough purple paint to cover 18 square meters. And now a solution is possible:

September 7, 2014September 14, 2014 | Puzzles