A problem from the 2000 Moscow Mathematical Olympiad:
Some of the cards in a deck are face down and some are face up. From time to time Pete draws out a group of one or more contiguous cards in which the first and last are both face down. He turns over this group as a unit and returns it to the deck in the position from which he drew it. Prove that eventually all the cards in the deck will be face up, no matter how Pete proceeds.
In 1831 this riddle appeared in a British publication titled Drawing Room Scrap Sheet No. 17:
In the morn when I rise, / I open my eyes, / Tho’ I ne’er sleep a wink all night;
If I wake e’er so soon, / I still lie till noon, / And pay no regard to the light.
I have loss, I have gain, / I have pleasure, and pain; / And am punished with many a stripe;
To diminish my woe, / I burn friend and foe, / And my evenings I end with a pipe.
I travel abroad. / And ne’er miss my road, / Unless I am met by a stranger;
If you come in my way, / Which you very well may, / You will always be subject to danger.
I am chaste, I am young, / I am lusty, and strong, / And my habits oft change in a day;
To court I ne’er go, / Am no lady nor beau, / Yet as frail and fantastic as they.
I live a short time, / I die in my prime, / Lamented by all who possess me;
If I add any more, / To what’s said before / I’m afraid you will easily guess me.
It was headed “For Which a Solution Is Required,” perhaps meaning that the editor himself did not know the solution. I think he may have found the riddle in The Lady’s Magazine, which had published it anonymously in September 1780 without giving the answer. Unfortunately he seems to have been disappointed — the Drawing Room Scrap Sheet never printed a solution either.
A century and a half later, in 1981, Faith Eckler challenged the readers of Word Ways: The Journal of Recreational Linguistics to think of an answer, offering a year’s subscription to the journal as a reward. When no one had claimed the prize by February 2010, Ross Eckler renewed his wife’s challenge, noting that the National Puzzlers’ League had also failed to find a solution.
That’s understandable — it’s tricky. “The author of the riddle cleverly uses ambiguous phrases to mislead the solver,” Ross Eckler notes. “I still lie till noon (inert, or continue to?); evenings I end with a pipe (a tobacco holder, or a thin reedy sound?); to court I ne’er go (a royal venue, a legal venue, or courtship?).”
To date, so far as I know, the riddle remains unsolved. Answers proposed by Word Ways readers have included fame, gossip, chessmen, a hot air balloon, and the Star and Stripes, though none of these seems beyond question. I offer it here for what it’s worth.
UPDATE: A solution has been found! Apparently The Lady’s Diary published the solution in 1783, which Ronnie Kon intrepidly ran to earth in the University of Illinois Rare Book and Manuscript Library. He published it in Word Ways in November 2012 (PDF). I’ll omit the solution here in case you’d still like to guess; be warned that it’s not particularly compelling. (Thanks, Ronnie.)
Starting in Delaware, you must tour the 48 contiguous United States, visiting each state exactly once.
Where will you finish?
A puzzle from Oswald Jacoby’s Mathematics for Pleasure (1962):
The MacDonalds are planning a long car journey of 27,000 miles. If they use tires that last 12,000 miles each, how many tires will they need, and how can they make the best use of them?
A quickie submitted by John Astolfi to MIT Technology Review’s Puzzle Corner, July/August 2013:
Consider the expansion of π (3.14159 …) in base 2. Does it contain more 0s than 1s, more 1s than 0s, or an equal number of both? Or is it impossible to tell?
In March 2013, New Mexico art dealer Forrest Fenn announced that he had hidden a bronze treasure chest in the Rocky Mountains north of Santa Fe. In the chest, he says, are gold coins, artifacts, and jewelry worth more than $1 million.
Fenn said he’d conceived the idea when diagnosed with cancer in 1988, planning to bury the treasure as a legacy. The cancer went into remission, but he decided to bury the chest anyway. In a self-published memoir he offered the following poem, which he says contains nine clues:
As I have gone alone in there
And with my treasures bold,
I can keep my secret where,
And hint of riches new and old.
Begin it where warm waters halt
And take it in the canyon down,
Not far, but too far to walk.
Put in below the home of Brown.
From there it’s no place for the meek,
The end is ever drawing nigh;
There’ll be no paddle up your creek,
Just heavy loads and water high.
If you’ve been wise and found the blaze,
Look quickly down, your quest to cease,
But tarry scant with marvel gaze,
Just take the chest and go in peace.
So why is it that I must go
And leave my trove for all to seek?
The answers I already know,
I’ve done it tired and now I’m weak.
So hear me all and listen good,
Your effort will be worth the cold.
If you are brave and in the wood
I give you title to the gold.
Fenn has been releasing further clues periodically as he follows the search (“No need to dig up the old outhouses, the treasure is not associated with any structure”). A number of people claim to have found the chest, but none has provided evidence, and Fenn says that to the best of his knowledge it remains undiscovered. There’s much background and discussion about the treasure at ttotc.com.
If you find it, I figure you owe me 75%.
A water jug is empty, and its center of gravity is above the inside bottom of the jug. Water is poured into the jug until the center of gravity of the jug and water (considered together) is as low as possible. Explain why this center of gravity must lie at the surface of the water.
A puzzle from Martin Gardner’s column in Math Horizons, November 1995:
Driving along the highway, Mr. Smith notices that signs for Flatz beer appear to be spaced at regular intervals along the roadway. He counts the number of signs he passes in one minute and finds that this number multiplied by 10 gives the car’s speed in miles per hour. Assuming that the signs are equally spaced, that the car’s speed is constant, and that the timed minute began and ended with the car midway between two signs, what is the distance from one sign to the next?