Here is a curious problem. We may safely assume that you had two parents; each of your parents had two parents, so that you had four grandparents. Arguing along similar lines you must have had eight great grandparents and so on. Assuming an average of three generations per century the number of your ancestors since the Christian Era began must have been nearly 1 trillion–
1,000,000,000,000,000,000 or 1018
This is vastly more people than have ever lived on the Earth. What can we do about it?
— J. Newton Friend, Numbers: Fun & Facts, 1954
This one is slippery, so watch it closely.
A poor old lady, with little money and plenty of time, sat quietly one day trying to devise a plan for making a little change. She finally came up with a very clever idea. Taking an old necklace, which she knew was worth only $4, she went to a pawnshop and pawned it for $3. Then, on a street corner, she started a friendly acquaintance with a young man, finally persuading him to buy the pawnticket for only $2. Now, she had $5 altogether and thus had made $1 profit. The pawnbroker wasn’t out any money since he paid only $3 for a $4 item, and the young man paid only $2 to get the $4 necklace. Who lost?
— Raymond F. Lausmann, Fun With Figures, 1965
From Lewis Carroll:
I don’t know if you are fond of puzzles, or not. If you are, try this. … A gentleman (a nobleman let us say, to make it more interesting) had a sitting-room with only one window in it–a square window, 3 feet high and 3 feet wide. Now he had weak eyes, and the window gave too much light, so (don’t you like ‘so’ in a story?) he sent for the builder, and told him to alter it, so as only to give half the light. Only, he was to keep it square–he was to keep it 3 feet high–and he was to keep it 3 feet wide. How did he do it? Remember, he wasn’t allowed to use curtains, or shutters, or coloured glass, or anything of that sort.
A printer prints a sentence in a monospaced font. It inserts a space after the concluding period and then prints the same sentence again. It continues in this way until it has filled the page, running the sentences together into one long paragraph. The sentence is shorter than a full line, and no words are hyphenated. Prove that the finished page will always include a full column of blank spaces.
A groaner from Clark Kinnaird’s Encyclopedia of Puzzles and Pastimes (1946):
“A farmer had 3 3/7 haystacks in one field and 5 4/9 haystacks in another field. He put them all together. How many did he have then?”
I’ll withhold the answer.
This is a story of four brothers. Billy owed a dollar to Jerry. Jerry owed a dollar to Tommy, and Tommy owed a dollar to Billy. The three of them met one day at a family picnic. Being brothers and good friends, none wished to hound the other about his debt. Vincent, the fourth brother, arrived at the picnic with some beer. While he was busily unloading the truck, Billy walked over, unnoticed, and quietly asked Vincent for a loan of a dollar, which Vincent gladly gave to him. Billy then ambled over to Jerry and paid him the dollar he owed him; then Jerry paid Tommy the dollar he owed to him; Tommy then went over to Billy and paid him the dollar he owed him. Billy then walked back to Vincent and paid him back his dollar. All old debts were paid. Simple, isn’t it?
— Raymond F. Lausmann, Fun With Figures, 1965
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?
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.
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?
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?
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.
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?
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?
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.
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
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.
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?
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.”
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?”