Counter Play

A devilish puzzle by Lee Sallows:

lee sallows counter play

In the diagram above, nine numbered counters occupy the cells of a 3×3 checkerboard so as to form a magic square. Any 3 counters lying in a straight line add up to 15. There are 8 of these collinear triads.

Reposition the counters (again, one to each cell) to yield 8 new collinear triads, but now showing a common sum of 16 rather than 15.

Husbands and Wives

This problem dates from at least 1774; this version appeared in the American Mathematical Monthly of December 1902:

Three Dutchmen and their wives went to market to buy hogs. The names of the men were Hans, Klaus, and Hendricks, and of the women, Gertrude, Anna, and Katrine; but it was not known which was the wife of each man. They each bought as many hogs as each man or woman paid shillings for each hog, and each man spent three guineas more than his wife. Hendricks bought 23 hogs more than Gertrude, and Klaus bought 11 more than Katrine. What was the name of each man’s wife?

(There are 21 shillings in a guinea.)

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Tight Squeeze

By T.R. Dowson. White to mate in 21 moves:

tight squeeze chess puzzle

It’s not as hard as it sounds, though it’s a bit like a square dance in a submarine.

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Reading Matter

In a certain library, no two books contain the same number of words, and the total number of books is greater than number of words in the largest book.

How many words does one of the books contain, and what is it about?

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Current Affairs

At 1:17 one afternoon a canoeist left his riverside camp and paddled upstream at 4 mph against a current of 1.5 mph. At 2:05 he passed a corked bottle floating downstream and noticed that it contained a message. He paddled some distance further but finally couldn’t help himself — he turned around and paddled after the bottle. He caught it just as it reached his camp. The message read:


“There is no reason why the camper should have paid any attention to this odd message, but you know how these things are,” writes Geoffrey Mott-Smith in Mathematical Puzzles for Beginners and Enthusiasts (1946). The camper had noticed a landmark at the point upstream where he’d turned around, so he was able to measure the distance the next day. But he could have reasoned the thing out from the facts. Can you?

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Clutch Cargo

Prove that the number of people who shake hands an odd number of times at the opera next Thursday will be even.

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Black and White


By W. Atkinson. White to mate in two moves.

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The Cigar Puzzle

From Henry Dudeney:

Two men are seated at a square-topped table. One places an ordinary cigar (flat at one end, pointed at the other) on the table, then the other does the same, and so on alternately, a condition being that no cigar shall touch another. Which player should succeed in placing the last cigar, assuming that they each will play in the best possible manner? The size of the table top and the size of the cigar are not given, but in order to exclude the ridiculous answer that the table might be so diminutive as only to take one cigar, we will say that the table must not be less than 2 feet square and the cigar not more than 4-1/2 inches long. With those restrictions you may take any dimensions you like. Of course we assume that all the cigars are exactly alike in every respect. Should the first player, or the second player, win?

Geoffrey Mott-Smith writes, “I cannot resist narrating that I first became acquainted with this gem while reading in bed, and that like an illustrious precursor I startled the household by jumping out of bed, dancing about crying ‘Eureka! Eureka!'”

What had he seen?

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The Potato Paradox


You have 100 pounds of Martian potatoes, which are 99 percent water by weight. You let them dehydrate until they’re 98 percent water. How much do they weigh now?

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A Chess Nut


By Sam Loyd. White to mate in two moves.

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The Handicap


Zachary challenges his brother Alexander to a 100-meter race. Alexander crosses the finish line when Zachary has covered only 97 meters.

The two agree to a second race, and this time Alexander starts 3 meters behind the starting line.

If both brothers run at the same speed as in the first race, who will win?

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Quitting Time

career rut alphametic

A depressing alphametic by Joseph Madachy. Each letter stands for a digit. What are the digits?

Chess and Dominoes

We learned in this problem that (spoiler!) if two squares of the same color are cut out of a chessboard, the remaining 62 squares cannot be tiled by 31 dominoes.

What if the squares removed are of different colors? Is the task possible then?

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economical chess puzzle

An anonymous puzzle from the British Chess Magazine, 1993. White to mate in half a move.

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“What Does The Mirror Say?”


The above line of figures does not appear very interesting at first sight, but if one asks some charming member of the fair sex to turn it upside down and hold it to a mirror to read it, a hidden meaning becomes apparent.

Strand, December 1908

Side Business


Here’s proof that one leg of a triangle always equals the sum of the other two.

ABC is our triangle. Extend it make a parallelogram, as shown, and divide the parallelogram into a grid. Obviously,

AB + BC = (AG + HJ + KL + MN) + (GH + JK + LM + NC).

Now let the grid grow increasingly fine: Instead of dividing the parallelogram into a 4×4 grid, make it 5×5, then 6×6, and so on. With each iteration, the stairstep figure described above will approximate AC more closely, and yet its total length will always equal AB + BC. Thus, at the limit, AB + BC = AC. Where is the error?

(From Henry Dudeney’s Canterbury Puzzles, via W.W. Rouse Ball’s Mathematical Recreations and Essays, 1892.)

The 65th Square

65th square

By Eric Angelini, Europe Echecs, 1990.

White adds one square at the edge of the board and then mates in two.

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“What Is Maggie’s Age?”

Another puzzle from Sam Loyd:

“How fast those children grow!” remarked Grandpa. “Tommy is now twice as old as Maggie was when Tommy was six years older than Maggie is now, and when Maggie is six years older than Tommy is now their combined ages will equal their mother’s age then, although she is now but forty-six.” How old is Maggie?

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A Remade Man


A tangram paradox by Sam Loyd. Each of these gentlemen is assembled from the same seven pieces. Yet one has a foot and the other doesn’t. How is this possible?

Car Hop


A classic railroad shunting puzzle. The segment at the top can accommodate either freight car, but not the locomotive. The freight cars can be joined if desired, and the locomotive can push or pull either or both cars from either direction. The task is to use the locomotive to swap the positions of the two cars.

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Track Meet

From Henry Dudeney:

A better class of puzzle is the well-known one of the Railway. If New York and San Francisco are just seven days’ journey apart, and if trains start from both ends every day at noon, how many trains coming in an opposite direction will a train leaving New York meet before it arrives at its destination at San Francisco?

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George Berkeley proposed that material things exist only because they are perceived to exist. In 1993, L.C. Rodó offered a suitably Berkeleyan puzzle in El Acertijo. Any piece, including a king, that is not regarded (attacked or guarded) by another piece disappears from the board. In the diagram above, every man is either attacked or guarded. But a single move (it may be by either White or Black) will set in motion a chain reaction in which all the pieces vanish from the board. What is the move?

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Assisted Suicide


From this position, how can White force Black to checkmate him in two moves?

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“The Caged Bird”

Very little is known about Theophilus Thompson, a chess problemist who was born a slave in Maryland in 1855. That’s a pity, because his work is often beautiful. Here’s a sample:

thompson chess puzzle

White to mate in two.

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