What is the product of this series?
(x – a) (x – b) (x – c) … (x – z)
This yields to an insight, so I’ll withhold the answer.
From J. Newton Friend, Numbers: Fun & Facts, 1954:
A boy jumped onto one end of a piece of tree trunk lying on the top of a hill. Now the log happened to be exactly 13 feet long, an unlucky omen for the youth, and the impact caused it to begin rolling down the hill. As it rolled, he managed to keep himself upright on top and slowly walked across the log to the other end which he reached just as the log came to rest at the bottom of the hill, 84 feet from where it began to roll.
The log was 2 feet in diameter. How far did the boy actually travel and how far would he have travelled had the log been 3 feet in diameter?
From a 1987 Hungarian math contest for 11-year-olds:
How can a 3 × 3 × 3 cube be divided into 20 cubes (not necessarily the same size)?
“I used to flatter myself that I would immediately be able to see through any problem that might be asked of an 11-year-old,” writes University of Waterloo mathematician Ross Honsberger. “I don’t take anything for granted anymore!”
Point A is the center of one square and the vertex of another. The side of each square is 4 inches. What is the area of the shaded region?
Here’s a long corridor with a moving walkway. Let’s race to the far end and back. We’ll both run at the same speed, but you run on the floor and I’ll run on the walkway, going “downstream” to the far end and “upstream” back to this point. Who will win?
This puzzle, by Les Marvin and Sherry Nolan, appeared in the Journal of Recreational Mathematics in 1977. “White to play in the adjoining diagram. If both players play optimally, will White win, lose, or draw?”
I don’t believe JRM ever published the solution. My stab: Either king is vulnerable to a check from the bishop file, and White will win a straight race. So I think Black must play defense. But if White attacks c7 with both knights and Black defends it doubly, then White can simply trade off all four knights (1. Nc7+ Nxc7 2. Nxc7+ Nxc7 bxc7) and the pawn will queen. So I think White wins.
This isn’t a very “mathematical” solution, but I can’t find a reliable alternative involving the parity of the knights’ moves, which seems to be what’s expected. Any ideas?
06/06/2014 UPDATE: A reader ran this position through a couple of strong chess engines and finds that it’s likely a draw — here’s one example:
[FEN “k6n/Pp4n1/1P6/8/8/6p1/1N4Pp/N6K w – – 0 1″]
1.Nd1 Nf7 2.Nc2 Ne5 3.Nce3 Nd7 4.Nd5 Nxb6 5.Nxb6+ Kxa7 6.Nc8+ Ka6 7.Ne3 b5 8.Nd6 b4 9.Ne4 Nh5 10.Nc2 Kb5 11.Nxb4 Kxb4 12.Nxg3 Nxg3+ 13.Kxh2 Nf1+ 14.Kh3 Ne3 15.g4 Kb3 16.g5 Nd5 17.g6 Nf4+ 18.Kg3 Nxg6
There doesn’t seem to be a sure way for either side to reach a win. I suspect that Marvin and Nolan thought otherwise, but they were writing in 1977, without the benefit of computer analysis. Without a published solution, we can’t be sure.
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?