Hot and Cold

Suppose you have three identical Dewar flasks labeled A, B, and C. (A Thermos is a Dewar flask.) You also have an empty container labeled D, which has thermally perfect conducting walls and which fits inside a Dewar flask.

Pour 1 liter of 80°C water into flask A and 1 liter of 20°C water into flask B. Now, using all four containers, is it possible to use the hot water to heat the cold water so that the final temperature of the cold water is higher than the final temperature of the hot water? How? (You can’t actually mix the hot water with the cold.)

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Pour half the cold water into D and put D into the hot flask (A). The final temperature in both A and D will be about 60°C. Empty D into C and the repeat the procedure, pouring the remaining half of the cold water in B into D and immersing it in the hot water. Now the final temperature of the water in A and D will be about 47°C. Empty D again into C; the final temperature of the mixture there will be about 53°C. So in two operations we’ve warmed the cold water to 53°C and cooled the hot water to 47°C. From Christopher Jargodzki’s books Science Brain-Twisters (1976) and Mad About Physics (2002).

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October 24, 2014November 2, 2014 | Puzzles · Science & Math

A Dice Puzzle

Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.

The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they’re different. Their chances of winning are equal.

The first die has 5 red faces and 1 blue face. What are the colors on the second die?

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Throwing two six-sided dice produces 36 possible outcomes. If Timothy and Urban have equal chances of winning, then there must be 18 outcomes in which both dice display the same color. If x is the number of red faces on the second die, then: 18 = 5x + 1(6 – x) x = 3 The second die must have 3 red faces and 3 blue faces. (From Pierre Berloquin, 100 Numerical Games, 1994.)

October 23, 2014November 3, 2014 | Puzzles · Science & Math

Black and White

Herlin chess problem

By Theodore Adrien Louis Herlin. White to mate in two moves.

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1. Qe8! From Schachzeitung der Berliner Schachgesellschaft, 1852.

October 18, 2014October 26, 2014 | Puzzles

Weighty Matters

From the Second All Soviet Union Mathematical Competition, Leningrad 1968:

On a teacher’s desk sits a balance scale, on which are a set of weights. On each weight is the name of at least one student. As each student enters the classroom, she moves all the weights that bear her name to the other side of the scale.

Before any students enter, the scale is tipped to the right. Prove that there’s some set of students that you can let into the room that will will tip the scale to the left.

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Make a list of all possible subsets of the students, including the empty set and the full set, and suppose you let in all of these groups. Each weight would spend half its time in the left pan and half in the right, so the total weight for each pan is the same. Since the empty set produces a tip to the right, there must be some other set that produces a tip to the left.

October 16, 2014October 26, 2014 | Puzzles

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