You have n cubical building blocks. You try to arrange them into the largest possible solid cube, but you find that don’t have quite enough blocks: One side of the large cube has exactly one row too few.
Prove that n is divisible by 6.
A problem from the 2003 Moscow Mathematical Olympiad:
A store has three floors, which are connected only by an elevator. At night the store is empty, and during the workday:
(1) Of the customers who enter the elevator on the second floor, half go to the first floor and half to the third floor.
(2) The number of customers who get out the elevator on the third floor is less than 1/3 the total number of customers who get out of the elevator.
Which is greater, the number of customers who go from the first floor to the second on a given workday, or the number who go from the first floor to the third?
Paul R. McClenon of Washington D.C. contributed this problem to the January-February 1964 issue of Recreational Mathematics Magazine:
The poor patient read the prescription which would save his life. ‘Mix carefully a one-pint drink, made of scotch whisky and water, mixed one to five (1/6 scotch, 5/6 water). Drink it quickly and go to bed.’
However, the patient finds only the following items at hand:
A one-quart bottle, about half full of scotch whisky.
An eight-ounce glass.
An unlimited supply of water from his faucet.
A sink with a drain.
No other containers or measuring devices.
He can pour from either container to the other, without spilling a drop, and can fill either to the brim without loss. How should he mix the prescription? Will he figure it out in time? Will he be saved? Did a doctor or bartender write this prescription?
The magazine went out of business before it could publish the solution. I’ll leave it to you.
05/17/2013 UPDATE: There seem to be a number of ways to accomplish this. Here’s one:
We need a 16-ounce dose that’s 1/6 whiskey, so the final mixture must contain 2.666 ounces of whiskey.
- Fill the 8-ounce glass with whiskey, then empty the jug.
- Return the glassful of whiskey to the empty jug and add two glassfuls of water.
- Fill the glass with this 2-to-1 mixture of water and whiskey. The 8-ounce glass now contains 2.666 ounces of whiskey, our target.
- Empty the bottle, pour the glass’ contents into it, and add one 8-ounce glassful of water.
That leaves us with 16 ounces in the jug, 1/6 of which is whiskey and the rest water, as directed.
Thanks to everyone who wrote in.
From The Youth’s Companion, Sept. 25, 1879:
Why is this man likely to succeed in life?
Why do we know he has reached middle life?
How does the picture indicate his occupation?
If you choose an answer to this question at random, what’s the chance that you’ll be correct?
(a) 25% (b) 50% (c) 0% (d) 25%
A 1936 poser by British puzzle maven Hubert Phillips:
A man I met in Fleet Street yesterday told me the following anecdote:
‘I met yesterday (he said) a friend of mine whom I had not seen since I was at Oxford. That was some years ago and we had not, during all that time, had any communication with one another. Nor had we at Oxford any friends or acquaintances in common.
‘I was delighted to see my friend, nevertheless. “I suppose,” I said, “that lots of things have happened to you.”
‘”Why, yes,” was the answer. “I am married now and this is my little girl.”
‘I looked at the child — a pretty little thing of about six. “And what is your name, my dear,” I asked her. “Margaret,” was the reply. “Aha,” said I, “the same name as your mother’s.”‘
How did the speaker know?
Bishop Samuel Wilberforce was fond of riddles. After his death in 1873, this one was found among his literary papers:
I’m the sweetest of sounds in Orchestra heard,
Yet in Orchestra never was seen.
I’m a bird of gay plumage, yet less like a bird,
Nothing ever in Nature was seen.
Touch the earth I expire, in water I die,
In air I lose breath, yet can swim and can fly;
Darkness destroys me, and light is my death,
And I only keep going by holding my breath.
If my name can’t be guessed by a boy or a man,
By a woman or girl it certainly can.
No one knows the answer.
A pilot is about to depart in his plane when he meets a young woman on the airport concourse. She has missed her flight.
“I can give you a lift if you like,” he says.
“But you don’t know where I’m going,” she says.
“It doesn’t matter. I can drop you off wherever you like and continue to my destination without going out of my way.”
This seems preposterous until he explains where he’s going. Where is it?
William West noticed this inscription in an alehouse near Brighton. What does it mean?
Sort the numbers 0, 1, 2, …, 123456 into two sets. In one set put all the numbers who digits add to an even sum; in the other put those whose digits produce an odd sum. Which set is larger?
A logic exercise by Lewis Carroll — what conclusion can be drawn from these premises?
- I despise anything that cannot be used as a bridge.
- Nothing that is worth writing an ode to would be an unwelcome gift to me.
- A rainbow will not bear the weight of a wheelbarrow.
- Whatever can be used as a bridge will bear the weight of a wheelbarrow.
- I would not take, as a gift, a thing that I despise.
Prove that the product of four consecutive positive integers cannot be a perfect square.
On Christmas Day 1877, assailed by two young ladies with “nothing to do,” Lewis Carroll invented a new “form of verbal torture”: Presented with two words of the same length, the solver must convert one to the other by changing a single letter at a time, with each step producing a valid English word. For example, HEAD can be converted to TAIL in five steps:
Carroll called the new pastime Doublets and published it in Vanity Fair, which hailed it as “so entirely novel and withal so interesting, that … the Doublets may be expected to become an occupation to the full as amusing as the guessing of the Double Acrostics has already proved.”
In some puzzles the number of steps is specified. In Nabokov’s Pale Fire, the narrator describes a friend who was addicted to “word golf.” “He would interrupt the flow of a prismatic conversation to indulge in this particular pastime, and naturally it would have been boorish of me to refuse playing with him. Some of my records are: HATE-LOVE in three, LASS-MALE in four, and LIVE-DEAD in five (with LEND in the middle).” I’ve been able to solve the first two of these fairly easily, but not the last.
But even without such a constraint, some transformations require a surprising number of steps. Carroll found that 10 were required to turn BLUE into PINK, and in 1968 wordplay expert Dmitri Borgmann declared himself unable to convert ABOVE into BELOW at all.
In a computer study of 5,757 five-letter English words, Donald Knuth found that most could be connected to one another, but 671 could not. One of these, fittingly, was ALOOF. In the wider English language, what proportion of words are “aloof,” words that cannot be connected to any of their fellows? Is ALOOF itself one of these?
In 1917 Sam Loyd and Thomas Edison made this short, which plays with similar ideas. The goat at the end was animated by Willis O’Brien, who would bring King Kong to life 16 years later: