The Magic Dice Cup

A tangram paradox from Sam Loyd’s Eighth Book of Tan (1903). Each of these cups was composed using the same seven geometric shapes. But the first cup is whole, and the others contain vacancies of different sizes.

“Of course it is a fallacy, a paradox, or an optical illusion, for you will say the feat is impossible!” But how is it done?

The Deliveryman’s Problem

A puzzle from L. Despiau’s Select Amusements in Philosophy and Mathematics, 1801:

Distribute among 3 persons 21 casks of wine, 7 of them full, 7 of them empty, and 7 of them half full, so that each of them shall have the same quantity of wine, and the same number of casks.

Click for Answer

Recalling Yesterday

From P.M.H. Kendall and G.M. Thomas, Mathematical Puzzles for the Connoisseur, 1962:

I’ve just been reading Jules Verne’s Around the World in Eighty Days — you know, where Phileas Fogg lost a day on the way round. Our science master says that ships put it right nowadays by having a thing called a Universal Date Line in the Pacific. When you cross the line from East to West you put the calendar on a day; and when you cross it the other way you put the calendar back. What I want to know is, when Puck put a girdle round the Earth in forty minutes and presumably did the right thing on crossing the Date Line, why didn’t he get back on the day before he started — or the day after, according to which way round he went?

I asked the English master this and he got quite cross about it and said it was nothing to do with Shakespeare. But if you flew round the earth as quickly as Puck it would matter, wouldn’t it?

Wouldn’t it? Why doesn’t Puck lose a day?

Click for Answer

Black and White

loyd american indian

By Sam Loyd. White to mate in two moves.

Click for Answer

The Annual Liars

Two brothers are scrupulously truthful, with one exception: Each lies about his birthday on his birthday.

On New Year’s Eve you ask what their birthdays are. The first says “Yesterday” and the second says “Tomorrow.”

On New Year’s Day you ask again what their birthdays are. Again the first says “Yesterday” and the second says “Tomorrow.”

What are their birthdays?

Click for Answer

A Chess Maze

nelson chess maze

Harry L. Nelson offered this puzzle in the Journal of Recreational Mathematics in 1983. The black king’s favorite square is c8, but he finds it is under attack by a white pawn. In how few moves can he correct this problem and return to a peaceful c8? White never moves. The black king can capture white pieces, but he may not visit any square more than once and may not enter check.

Click for Answer

A Thought Experiment


Suppose you have two identical bolts. Hold each by its head, engage the threads as shown, and revolve one about the other. Will this action pull the heads closer together or drive them farther apart?

Click for Answer

Black and White

sam loyd mate in two

By Sam Loyd. White to mate in two moves.

Click for Answer

Black and White

Sam Loyd chess puzzle, Detroit Free Press, 01-27-1877

By Sam Loyd. White to mate in two moves.

Click for Answer

A Pretty Problem

In Longfellow’s novel Kavanagh, Mr. Churchill reads a word problem to his wife:

“In a lake the bud of a water-lily was observed, one span above the water, and when moved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

“That is charming, but must be very difficult,” she says. “I could not answer it.”

Is it? If a span is 9 inches and a cubit is 18 inches, how deep is the water?

Click for Answer


For 25 years, Macalester College mathematician Joe Konhauser offered a “problem of the week” to his students. Here’s a sample, from the collection Which Way Did the Bicycle Go? (1996):

Fifteen sheets of paper of various sizes and shapes lie on a desktop, covering it completely. The sheets may overlap one another and may even hang over the edge of the desktop. Prove that five of the sheets can be removed so that the remaining ten sheets cover at least two-thirds of the desktop.

Click for Answer

Even Steven

A Scholar traveyling, and having noe money, call’d at an Alehouse, and ask’d for a penny loafe, then gave his hostesse it againe, for a pot of ale; and having drunke it of, was going away. The woman demanded a penny of him. For what? saies he. Shee answers, for ye ale. Quoth hee, I gave you ye loafe for it. Then, said she, pay for ye loafe. Quoth hee, had you it not againe? which put ye woman to a non plus, that ye scholar went free away.

— John Ashton, Humour, Wit, & Satire of the Seventeenth Century, 1883

Trial by Fire

You are grilling steaks for Genghis Khan. Your little grill can broil two steaks at a time, but Genghis is hungry and wants three. That’s a problem: It takes 4 minutes to grill each side of a steak, so you’ll spend 8 minutes grilling the first two steaks, then another 8 grilling the third. Sixteen minutes is a long time to keep a warlord waiting.

How can you improve your time while still cooking the steaks thoroughly? Genghis really likes his well done.

Click for Answer


do-si-do 4 x 3 chess puzzle

I’m not sure who originated this puzzle. Can the white queen force the black king onto square a1 of this 4 x 3 chessboard?

Click for Answer

Round and Round


Does the top of a rolling wheel move faster than the bottom? In his Cyclopedia of Puzzles (1914), Sam Loyd calls this “an old problem which has created a considerable discussion in the mechanical world.”

The rolling wheel retains its shape; it will arrive at its destination as a connected unit. This seems to imply that all of its parts are moving at the same speed. Yet the point in contact with the ground is moving not at all, while the top continuously overtakes it. Surely, then, the top is moving faster? “There is just enough of the mathematical and mechanical element in the make-up of the problem,” writes Loyd, “to provoke discussions from such as are well-up on these subjects.”

His answer: “The top of a wheel progresses exactly as fast as the bottom.” And, being Sam Loyd, he adds a wrinkle: “If the question referred to a mark on the tire the answer would be different, for the top is the highest point of the wheel and cannot revolve, for if it revolves the hundredth part of an inch it ceases to be the top.”

A second vexed wheel riddle; William James and Lewis Carroll consider related questions; a train moves simultaneously east and west.

Meant to Be

In 2007 Gordon Bonnet was going through some genealogical records when he ran across a marriage record for Edward DeVere Stewart and Etta Grace Staggers.

“It was only when I was putting the names in my database that I noticed something odd about them. What is it?”

Click for Answer



By Sam Loyd. White to move and mate in two.

Click for Answer

Long Range

A puzzler from Willis Ernest Johnson, Mathematical Geography, 1907:

“A man was forty rods due east of a bear, his gun would carry only thirty rods, yet with no change of position he shot and killed the bear. Where on earth were they?”

Click for Answer

The Missing Footmen

Like Nabokov, Lord Dunsany was fond of composing chess problems. This example was published in Hubert Phillips’ Week-end Problems Book of 1932:

dunsany - the missing footmen chess problem - 1

Two eccentric gentlemen abandoned this position at a chess club — White had announced mate in four. What was the mate?

The key is to notice that Black’s king and queen have changed positions. This is not possible if the black pawns are on their home squares. And this means that we’ve been viewing the position upside down:

dunsany - the missing footmen chess problem - 2

Now it’s clear that Black is hemmed in by his own pawns — White can mate him in four moves with the b8 knight, e.g. by 1. Nc6 Nf3 2. Nb4 (threatening Nd3#) Ne5 3. Qxe5 (any) 4. Nd3#.

Dunsany drew a game against Capablanca in a simultaneous exhibition in London in April 1929. He later wrote:

One art they say is of no use;
The mellow evenings spent at chess,
The thrill, the triumph, and the truce
To every care, are valueless.

And yet, if all whose hopes were set
On harming man played chess instead,
We should have cities standing yet
Which now are dust upon the dead.

Gender Issues

In a certain kingdom, boys and girls are born in strictly equal proportions. Determined to increase the proportion of women in the land, the sultan issues a decree: Any woman who bears a son is forbidden to have any further children. He reasons that some families will thus contain multiple daughters but a single son.

A number of years pass, and the sultan is confused to find that the kingdom still contains an equal number of boys and girls. Why?

Click for Answer

The Motor Hire

A puzzle from R.M. Abraham, Diversions & Pastimes, 1933:

Michael O’Bleary hired a motor-car at a cost of fifteen dollars to take him to Ballygoogly market and back again in the evening. When he got half-way on his outward journey he met a friend, gave him a lift to the market, and brought him back to the point where he picked him up in the morning. There was a dispute about the payment. How much should Michael charge his passenger for his share of the motor hire?

Click for Answer

Black and White

sam loyd chess puzzle 285

By Sam Loyd. White to mate in two.

Click for Answer



You must participate in a three-way duel with two rivals. Each of you is given a pistol and unlimited ammunition. Unfortunately, you, Red, are the weakest shot — you hit your target only 1/3 of the time. Black is successful 2/3 of the time, and Gray hits everything he aims at.

It’s agreed that you will take turns: You’ll shoot first, then Black, then Gray, and you’ll continue in this order until one survivor remains. At whom should you shoot?

Click for Answer

Adding Up

What is the sum of all the figures in the numbers from 1 to 1 million?

Hint: With the right technique, this can be done in the head.

Click for Answer