Babylas, Hilary, and Sosthenes have escaped the tower and divided their treasure into three bags. But now they must cross a river, and the boat can accommodate only two men at a time, or one man and a bag. None will trust another with his bag on the shore, but they agree that a man in the boat can be trusted to drop or retrieve a bag at either shore, as he’ll be too busy to tamper with it. How can they cross the river?
Three robbers, Babylas, Hilary, and Sosthenes, are stealing a treasure chest from the top of an old tower. Unfortunately, they’ve had to destroy their ladder to avoid pursuit, so they’ll have to descend using a crude tackle — a single pulley and a long rope with a basket at each end.
Babylas weighs 170 pounds, Hilary 100 pounds, Sosthenes 80 pounds, and the treasure 60 pounds. If the difference in weight between the two baskets is greater than 20 pounds then the heavier basket will descend too quickly and injure its occupant (though the treasure chest can withstand this). How can the three of them safely escape the tower with the treasure?
Upend Congress and pour its members onto Constitution Avenue. Each member has up to three enemies. Prove that it’s possible to pack the 535 members back into the House and Senate in such a way that none of them has more than one enemy in his chamber. (Enmity is always mutual — I am my enemy’s enemy — and the chambers need not retain their former sizes.)
Ten weary, footsore travelers,
All in a woeful plight,
Sought shelter at a wayside inn
One dark and stormy night.
“Nine beds — no more,” the landlord said,
“Have I to offer you;
To each of eight a single room,
But the ninth must serve for two.”
A din arose. The troubled host
Could only scratch his head,
For of those tired men, not two
Could occupy one bed.
The puzzled host was soon at ease —
He was a clever man —
And so to please his guests devised
This most ingenious plan.
In room marked A, two men were placed,
The third he lodged in B,
The fourth to C was then assigned —
The fifth retired to D.
In E the sixth he tucked away,
In F the seventh man;
The eighth and ninth in G and H,
And then to A he ran,
Wherein the host, as I have said,
Had laid two travelers by;
Then taking one, the tenth and last,
He lodged him safe in I.
Nine single rooms — a room for each —
Were made to serve for ten.
And this it is that puzzles me
And many wiser men.
— S.R. Ford, Ford’s Christian Repository & Home Circle, May 1888
A puzzle from Lewis Carroll’s diary:
The Dodo says that the Hatter tells lies.
The Hatter says that the March Hare tells lies.
The March Hare says that both the Dodo and the Hatter tell lies.
Who is telling the truth?
A riddle by Jonathan Swift:
We are little airy creatures,
All of different voice and features:
One of us in glass is set,
One of us you’ll find in jet,
T’other you may see in tin,
And the fourth a box within;
If the fifth you should pursue,
It can never fly from you.
What are we?
A problem from the Soviet Mathematical Olympiad:
Two hundred students are arranged in 10 rows of 20 children. The shortest student in each column is identified, and the tallest of these is marked A. The tallest student in each row is identified, and the shortest of these is marked B. If A and B are different people, which is taller?
Put the integers 1, 2, 3, … n in any order and call them a1, a2, a3, … an. Then form the product
P = (a1 – 1) × (a2 – 2) × (a3 – 3) … × (an – n).
Now: If n is odd, prove that P is even.
Two men are brothers-in-law if one is married to the other’s sister. What is the largest possible group of men in which each is brother-in-law to each of the others?
An ant will always position itself so that it’s precisely twice as far from vinegar as from honey. If we put a dab of vinegar at A and a dab of honey at B and we release a troop of ants, what formation will they take up?
A.J. Fink, Good Companions, February 1917. White to mate in two moves.
From Miscellanea Curiosa: or, Entertainments for the Ingenious of Both Sexes, January 1734:
One evening, as I walk’d to take the Air,
I chanc’d to overtake two Ladies fair;
Each by the Hand a lovely Boy did lead,
To whom in courteous Manner thus I said:
Ladies! so far oblige me as to shew
How near akin these Boys are unto you?
They, smiling, quickly made this dark Reply,
Sons to our Sons they are, we can’t deny:
Though it seem strange, they are our Husbands’ Brothers,
And likewise each is Uncle to the other:
They both begot, and born in Wedlock were,
And we their Mothers and Grandmothers are.
Now try if you this Mystery can declare.
Will a prime number ever appear in this series?
A problem by Hungarian mathematician Laszlo Lovász:
A track has n arbitrarily spaced fuel depots. Each depot contains a quantity of gasoline; the total amount of gas is exactly enough to take us around the track once. Prove that, no matter how the gas is distributed, there will be a depot at which an empty car can fill up, proceed around the track picking up gas at each depot, and complete a full round trip back to its starting depot.
This problem originated in Russia, according to various sources, but no one’s sure precisely where:
Before you is a square table that can rotate freely. In each corner is a deep well, at the bottom of which is a tumbler that’s either upright or inverted. You can’t see the tumblers, but you can reach into the wells to feel their positions.
Periodically the table rotates and stops at random. After each stop, you can feel two of the tumblers and turn over either, both, or neither. If all four of the tumblers are in the same state — all upright or all inverted — then a bell sounds. Otherwise the table rotates again and you make another “move.”
Can you guarantee to ring the bell in a finite number of moves? If so, how?
A logic puzzle by Lewis Carroll, July 2, 1893. What conclusion can be drawn from these premises?
- All who neither dance on tight-ropes nor eat penny-buns are old.
- Pigs that are liable to giddiness are treated with respect.
- A wise balloonist takes an umbrella with him.
- No one ought to lunch in public who looks ridiculous and eats penny-buns.
- Young creatures who go up in balloons are liable to giddiness.
- Fat creatures who look ridiculous may lunch in public, provided that they do not dance on tight-ropes.
- No wise creatures dance on tight-ropes if liable to giddiness.
- A pig looks ridiculous carrying an umbrella.
- All who do not dance on tight-ropes and who are treated with respect are fat.
Otto Wurzburg, first prize, eighth tourney, Pittsburgh Gazette-Times, June 10, 1917. White to mate in two.
Stephen Barr observes that a pitched roof receives less rain per unit area than level ground does. This seems to mean that rain that falls at a slant will be less wetting than rain that falls vertically. Why isn’t this so?
From the U.K. Schools Mathematical Challenge, a multiple-choice competition for students ages 11-14:
Humphrey the horse at full stretch is hard to match. But that is just what you have to do: move one match to make another horse just like (i.e. congruent to) Humphrey. Which match must you move?
A ladder is leaning against a tree. On the center rung is a pussycat. She must be a very determined pussycat, because she remains on that rung as we draw the foot of the ladder away from the tree until the ladder is lying flat on the ground. What path does the pussycat describe as she undergoes this indignity?