Futility Closet book ad


On Target

Samuel Isaac Jones offered this poser in his 1929 book Mathematical Wrinkles:

Cook was within 10 miles of the north pole and Peary was also within 10 miles of the pole, but 20 miles from Cook. What direction was Peary from Cook? Suppose Peary threw a ball at Cook and hit him. In what direction did the ball go?

He omitted the answer, apparently inadvertently. What is it?

A Missed Opportunity

bruno sommer missed opportunity

By Bruno Sommer, 1910. White has just moved, and he realizes too late that he could have mated Black on the move. What was his last move, and what was the mate?

Click for Answer

The Crossing

A family of four has to cross a river. The father and mother each weigh 150 pounds, and each of the two sons weighs 75 pounds. Unfortunately, the boat will carry only 150 pounds maximum. How can they get across?

Click for Answer

The Last Redoubt


By Sam Loyd. White to mate in two.

Click for Answer

“The Best Bridge Problem Ever Invented”


I don’t play bridge, so I’m posting this somewhat blindly. It was devised by W.H. Whitfeld, card editor of the Field, apparently in the late 19th century. The reader who submitted it to the Strand wrote, “If you don’t know the solution, I guarantee that it will take you or any of your staff three or four days.”

“We have a higher opinion of our readers’ skill than to allot them such a time-limit as this,” wrote the editors. “But certainly anyone who can solve this problem in three or four hours will have good cause to be congratulated on his ingenuity.”

Click for Answer

A French Cryptogram

When the Chevalier de Rohan was sent to the Bastille in 1674 on suspicion of treason, he knew there was no evidence against him except what might be extracted from one other prisoner. His friends had promised to communicate the result of that examination, and in sending him some fresh clothing they wrote on one of the shirts MG DULHXCCLGU GHJ YXUJ, LM CT ULGC ALJ.

For 24 hours de Rohan puzzled over the message, but he could make no sense of it. Despairing, he admitted his guilt and was executed. What was the message?

Click for Answer

“The Hidden Star”

dudeney - hidden star

From Henry Dudeney:

The ilustration represents a square tablecloth of choice silk patchwork. This was put together by the members of a family as a little birthday present for one of its number. One of the contributors supplied a portion in the form of a perfectly symmetrical star, and this has been worked in exactly as it was received. But the triangular pieces so confuse the eye that it is quite a puzzle to find the hidden star.

Can you discover it, so that, if you wished, by merely picking out the stitches, you could extract it from the other portions of the patchwork?

Breaking Bad


Amy and Betty are playing a game. They have a chocolate bar that’s 8 squares long and 6 squares wide. Amy begins by breaking the bar in two along any division. Betty can then pick up any piece and break it in two, and so on. The first player who cannot move will be clapped in chains and rocketed off to a lifetime of soul-destroying toil in the cobalt mines of Yongar Zeta. (I know, it’s a pretty brutal game.) Who will win?

Click for Answer

The Kitchen Snitch

A logic puzzle from Mathematical Circles (Russian Experience), a collection of problems for Soviet high school math students:

During a trial in Wonderland the March Hare claimed that the cookies were stolen by the Mad Hatter. Then the Mad Hatter and the Dormouse gave testimonies which, for some reason, were not recorded. Later on in the trial it was found out that the cookies were stolen by only one of these three defendants, and, moreover, only the guilty one gave true testimony. Who stole the cookies?

Click for Answer

Cash and Carry

A favorite problem of Lewis Carroll involves a customer trying to complete a purchase using pre-decimal currency. He wants to buy 7s. 3d. worth of goods, but he has only a half-sovereign (10s.), a florin (2s.), and a sixpence. The shopkeeper can’t give him change, as he himself has only a crown (5s.), a shilling, and a penny. As they’re puzzling over this a friend enters the shop with a double-florin (4s.), a half-crown (2s. 6d.), a fourpenny-bit, and a threepenny-bit. Can the three of them negotiate the transaction?

Happily, they can. They pool their money on the counter, and the shopkeeper takes the half-sovereign, the sixpence, the half-crown, and the fourpenny-bit; the customer takes the double-florin, the shilling, and threepenny-bit as change; and the friend takes the florin, the crown, and the penny.

“There are other combinations,” writes John Fisher in The Magic of Lewis Carroll, “but this is the most logistically pleasing, as it will be seen that not one of the three persons retains any one of his own coins.”

Related: From Henry Dudeney, a magic square:


(Strand, December 1896)

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.

A Geometric Paradox

a geometric paradox

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.)

Click for Answer

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.

Click for Answer

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?

Click for Answer

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?

Click for Answer

Clutch Cargo

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

Click for Answer

Black and White


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

Click for Answer

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?

Click for Answer

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?

Click for Answer

A Chess Nut


By Sam Loyd. White to mate in two moves.

Click for Answer

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?

Click for Answer

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?

Click for Answer