## Looking Up

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.

## What Am I?

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.)

## End State

Starting in Delaware, you must tour the 48 contiguous United States, visiting each state exactly once.

Where will you finish?

## Another Car Puzzle

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?

## Portions of Pi

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?

## The Fenn Treasure

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%.

## Falling Gravity

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.

## Time and Distance

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?

## Lineup

A group of children are standing outside a room. Each wears a hat that’s either red or blue, and each child can see the other children’s hats but not her own. At a signal they enter the room one by one and arrange themselves in a line partitioned by hat color. How do they manage this without communicating?

## Book Codes

Benedict Arnold encrypted his messages to the British Army using Blackstone’s *Commentaries on the Laws of England*. Arnold would replace each word in his message with a triplet of numbers representing the page number, line number, and word position where the word might be found in Blackstone. For example:

The 166.8.11 of the 191.9.16 are 129.19.21 266.9.14 of the .286.8.20, and 291.8.27 to be on 163.9.4 115.8.16 114.8.25ing — 263.9.14 are 207.8.17ed 125.8.15 103.8.60 from this 294.8.50 104.9.26 — If 84.8.9ed — 294.9.12 129.8.7 only to 193.8.3 and the 64.9.5 290.9.20 245.8.3 be at an 99.8.14.

British Army Major John André could then look up the words in his own copy of Blackstone to discover Arnold’s meaning:

The mass of the People are heartily tired of the War, and wish to be on their former footing — They are promised great events from this year’s exertion — If disappointed — you have only to persevere and the contest will soon be at an end.

The danger in using a book code is that the enemy can decode the messages if he can identify the book — and sometimes even if he can’t. In the comic strip *Steve Roper*, a reporter once excitedly telephoned the coded message 188-1-22 71-2-13 70-2-11 68-1-25 19-1-6 112-2-10 99-1-35. Reader Sean Reddick suspected that this message had been encoded using a dictionary, with each triplet of numbers denoting page, column, and word number. He never did discover the book that had been used, but by considering the ratios involved and consulting half a dozen dictionaries he managed to break the code anyway — he sent his solution to a nationally known columnist, who verified his feat when the comic strip bore out his solution. What was the message? (Hint: In the comic, the reporter mentions significantly that the plaintext message was given to him by “the delivery boy.”)

## One Two Three

Each point on a straight line is either red or blue. Show that it’s always possible to find three points of the same color in which one is the midpoint of the other two.

## There and Back Again

John and Mary drive from Westville to Eastville. John drives the first 40 miles, and Mary drives the rest of the way. That afternoon they return by the same route, with John driving the first leg and Mary driving the last 50 miles. Who drives the farthest, and by what distance?

## The Sixth Cent

You toss 6 fair coins, and I toss 5 fair coins. What is the probability that you get more heads than I do?

## The Three Hats Game

Three players enter a room, and a maroon or orange hat is placed on each one’s head. The color of each hat is determined by a coin toss, and the outcome of one toss has no effect on the others. Each player can see the other players’ hats but not his own.

The players can discuss strategy before the game begins, but after this they may not communicate. Each player considers the colors of the other players’ hats, and then simultaneously each player must either guess the color of his own hat or pass.

The group shares a $3 million prize if at least one player guesses correctly and no player guesses incorrectly. What strategy will raise their chance of winning above 50 percent?

## Red and Black

I propose a card game. I’ll shuffle an ordinary deck of cards and turn up the cards in pairs. If both cards in a given pair are black, I’ll give them to you. If both are red, I’ll take them. And if one is black and one red, then we’ll put them aside, belonging to no one.

You pay a dollar for the privilege of playing the game, and then we’ll go through the whole deck. When the game is over, if you have no more cards than I do, you pay nothing. But for every card that you have more than I, I’ll pay you 3 dollars. Should you play this game?

## Black and White

By W.A. Shinkman. This is a self-mate in two moves: White makes a move, Black is allowed to make any legal reply, then White plays a second move that forces Black to checkmate him.

## A Holiday Puzzle

New Year’s Day normally falls one week after Christmas: If Christmas falls on a Thursday, then New Year’s Day will fall on a Thursday as well. What is the most recent year in which Christmas and New Year’s Day fell on different days of the week?

## Switching Visits

A prison warden greets 23 new prisoners with this challenge. They can meet now to plan a strategy, but then they’ll be placed in separate cells, with no means of communicating. Then the warden will take the prisoners one at a time to a room that contains two switches. Each switch has two positions, on and off, but they’re not connected to anything. The prisoners don’t know the initial positions of the switches. When a prisoner is led into the room, he must reverse the position of exactly one switch. Then he will be led back to his cell, and the switches will remain undisturbed until the warden brings the next prisoner. The warden chooses prisoners at his whim, and he may even choose one prisoner several times in a row, but at any time, each prisoner is guaranteed another visit to the switch room.

The warden continues doing this until a prisoner tells him, “We have all visited the switch room.” If this prisoner is right, then all the prisoner will be set free. But if he’s wrong then they’ll all remain prisoners for life.

What strategy can the prisoners use to ensure their freedom?

## Threewise

Draw an arbitrary triangle and build an equilateral triangle on each of its sides, as shown.

Now show that AP = BQ = CR.