Times Square

Prove that the product of four consecutive positive integers cannot be a perfect square.

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If n is the smallest of the four integers, then the product is (n)(n+1)(n+2)(n+3) = (n² + 3n)(n² + 3n + 2) = (n² + 3n + 1)² – 1 This can’t be a perfect square, because two positive squares cannot differ by 1. From Angela Dunn’s Mathematical Bafflers, via David Wells’ Penguin Book of Curious and Interesting Puzzles, 1992.

March 13, 2013March 22, 2013 | Puzzles · Science & Math

Word Ladders

On Christmas Day 1877, assailed by two young ladies with “nothing to do,” Lewis Carroll invented a new “form of verbal torture”: Presented with two words of the same length, the solver must convert one to the other by changing a single letter at a time, with each step producing a valid English word. For example, HEAD can be converted to TAIL in five steps:

HEAD
HEAL
TEAL
TELL
TALL
TAIL

Carroll called the new pastime Doublets and published it in Vanity Fair, which hailed it as “so entirely novel and withal so interesting, that … the Doublets may be expected to become an occupation to the full as amusing as the guessing of the Double Acrostics has already proved.”

In some puzzles the number of steps is specified. In Nabokov’s Pale Fire, the narrator describes a friend who was addicted to “word golf.” “He would interrupt the flow of a prismatic conversation to indulge in this particular pastime, and naturally it would have been boorish of me to refuse playing with him. Some of my records are: HATE-LOVE in three, LASS-MALE in four, and LIVE-DEAD in five (with LEND in the middle).” I’ve been able to solve the first two of these fairly easily, but not the last.

But even without such a constraint, some transformations require a surprising number of steps. Carroll found that 10 were required to turn BLUE into PINK, and in 1968 wordplay expert Dmitri Borgmann declared himself unable to convert ABOVE into BELOW at all.

In a computer study of 5,757 five-letter English words, Donald Knuth found that most could be connected to one another, but 671 could not. One of these, fittingly, was ALOOF. In the wider English language, what proportion of words are “aloof,” words that cannot be connected to any of their fellows? Is ALOOF itself one of these?

In 1917 Sam Loyd and Thomas Edison made this short, which plays with similar ideas. The goat at the end was animated by Willis O’Brien, who would bring King Kong to life 16 years later:

March 11, 2013September 28, 2019 | Language · Puzzles

Black and White

kockelkorn chess problem

By Carl Kockelkorn and Johannes Kohtz, 1876. White to mate in two moves.

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The key is to promote the pawn to a bishop: 1. f8=B Kxe8 2. Qc8#. Only a bishop will do — queening the pawn gives stalemate.

March 8, 2013March 15, 2013 | Puzzles

Ends and Means

Prove that if each point in the plane is colored red, yellow, or blue, a unit segment must exist whose endpoints are the same color.

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Set up two abutting equilateral triangles ABC and BCD with unit sides, as above, and suppose A is red. In order to avoid a unit segment with similarly colored endpoints, D must also be red. By the same reasoning, every point at this distance from A must also be red. But this describes a monochromatic circle, and we can always find two points on this circle that are 1 unit apart. Thus it’s impossible to avoid having a unit segment with ends of the same color. From Joseph Konhauser, Macalester College, St. Paul., Minn., in the Pi Mu Epsilon Journal, 1984, 668, via Ross Honsberger, Mathematical Chestnuts From Around the World, 2001.

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March 5, 2013March 15, 2013 | Puzzles

Black and White

shinkman chess problem

By William Anthony Shinkman. White to mate in two moves.

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White wins with the absurd-looking 1. Qe6! Every Black response leads to mate. From Deutsche Schachzeitung, 1875.

March 1, 2013March 8, 2013 | Puzzles

In the Dark

From The Black-Out Book, a book of entertainments to amuse Londoners during the air-raid blackouts of 1939:

When a certain commodity was rationed, a woman wrote to her tradesman asking his advice.

He replied:

If the B mt, put :
If the B . putting :

What was the commodity, and what did he mean?

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The tradesman was a coal merchant, and his message meant: If the grate be empty put coal on. If the grate be full stop putting coal on.

February 28, 2013March 8, 2013 | Puzzles

Black and White

wheeler chess problem

By Charles Henry Wheeler. White to mate in two moves.

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Only 1. Ra1!, withdrawing the rook to a guarded square, leads to mate in every variation. From Chess Problems, 1880.

February 22, 2013April 24, 2013 | Puzzles

Black and White

de boer chess problem

By Gerke Luiten De Boer. White to mate in two moves.

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White wins with the most unlikely move on the board, 1. Nd4!, threatening to jump to b5 or f5 with mate. Black can stop this only by capturing the knight with a pawn, which permits White to push a pawn of his own, giving mate. From Sissa, 1872.

February 15, 2013April 29, 2013 | Puzzles

Well Done

Using a 7-quart and a 3-quart jug, how can you obtain exactly 5 quarts of water from a well?

That’s a water-fetching puzzle, a familiar task in puzzle books. Most such problems can be solved fairly easily using intuition or trial and error, but in Scripta Mathematica, March 1948, H.D. Grossman describes an ingenious way to generate a solution geometrically.

Let a and b be the sizes of the jugs, in quarts, and c be the number of quarts that we’re seeking. Here, a = 7, b = 3, and c = 5. (a and b must be positive integers, relatively prime, where a is greater than b and their sum is greater than c; otherwise the problem is unsolvable, trivial, or can be reduced to smaller integers.)

Using a field of lattice points (or an actual pegboard), let O be the point (0, 0) and P be the point (b, a) (here, 3, 7). Connect these with OP. Then draw a zigzag line Z to the right of OP, connecting lattice points and staying as close as possible to OP. Now “It may be proved that the horizontal distances from OP to the lattice-points on Z (except O and P) are in some order without repetition 1, 2, 3, …, a + b – 1, if we count each horizontal lattice-unit as the distance a.” In this example, if we take the distance between any two neighboring lattice points as 7, then each of the points on the zigzag line Z will be some unique integer distance horizontally from the diagonal line OP. Find the one whose distance is c (here, 5), the number of quarts that we want to retrieve.

Now we have a map showing how to conduct our pourings. Starting from O and following the zigzag line to C:

Each horizontal unit means “Pour the contents of the a-quart jug, if any, into the b-quart jug; then fill the a-quart jug from the well.”
Each vertical unit means “Fill the b-quart jug from the a-quart jug; then empty the b-quart jug.”

So, in our example, the map instructs us to:

Fill the 7-quart jug.
Fill the 3-quart jug twice from the 7-quart jug, each time emptying its contents into the well. This leaves 1 quart in the 7-quart jug.
Pour this 1 quart into the 3-quart jug and fill the 7-quart jug again from the well.
Fill the remainder of the 3-quart jug (2 quarts) from the 7-quart jug and empty the 3-quart jug. This leaves 5 quarts in the 7-quart jug, which was our goal.

You can find an alternate solution by drawing a second zigzag line to the left of OP. In reading this solution, we swap the roles of a and b given above, so the map tells us to fill the 3-quart jug three times successively and empty it each time into the 7-quart jug (leaving 2 quarts in the 3-quart jug the final time), then empty the 7-quart jug, transfer the remaining 2 quarts to it, and add a final 3 quarts. “There are always exactly two solutions which are in a sense complementary to each other.”

Grossman gives a rigorous algebraic solution in “A Generalization of the Water-Fetching Puzzle,” American Mathematical Monthly 47:6 (June-July 1940), 374-375.

February 14, 2013April 27, 2022 | Puzzles · Science & Math

Clatter Plot

A puzzle by Lewis Carroll:

Two travelers, starting at the same time, went opposite ways round a circular railway. Trains start each way every 15 minutes, the easterly ones going round in 3 hours, the westerly in 2. How many trains did each meet on the way, not counting trains met at the terminus itself?

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The trains one way took 180 minutes, the other way 120. Let us take the l.c.m., 360, and divide the railway into 360 units. Then one set of trains went at the rate of 2 units a minute and at intervals of 30 units; the other at the rate of 3 units a minute and at intervals of 45 units. An easterly train starting has 45 units between it and the first train it will meet: it does 2/5 of this while the other does 3/5, and thus meets it at the end of 18 units, and so all the way round. A westerly train starting has 30 units between it and the first train it will meet: it does 3/5 of this while the other does 2/5, and thus meets it at the end of 18 units, and so all the way round. Hence if the railway be divided, by 19 posts, into 20 parts, each containing 18 units, trains meet at every post, and each traveler passes 19 posts in going round, and so meets 19 trains. From A Tangled Tale, 1886.

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February 13, 2013May 1, 2013 | Puzzles