All Aboard

One hundred people board a 100-seat airplane. The first one has lost his boarding pass, so he sits in a random seat. Each subsequent passenger sits in his own seat if it’s available or takes a random unoccupied seat if it’s not.

What’s the probability that the 100th passenger finds his seat occupied?

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It’s 50 percent. When the 100th passenger boards, the last available seat must be either the one assigned to the first passenger or the one assigned to the last. Nothing favors either of these outcomes; they’re equally likely.

February 29, 2012January 31, 2013 | Puzzles

Making Ends Meet

making ends meet puzzle

Fifty coins of various denominations are arranged in a row. You will take a coin from either of the row’s ends, then I will, and so on until all the coins are gone. What strategy will ensure that you take at least as much money as I?

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Before you begin, number the coins from 1 to 50 and determine whether the even-numbered or odd-numbered coins yield a higher sum. Because you’re moving first, you can contrive to capture all of those coins. For example, if the even-numbered coins are more valuable, begin by taking coin 50. That forces me to take 1 or 49, which permits you to take 2 or 48, and so on. From Peter Winkler’s excellent book Mathematical Puzzles: A Connoisseur’s Collection, 2004.

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February 27, 2012January 31, 2013 | Puzzles

Half and Half

half and half puzzle

It’s easy to draw a line and scatter an equal number of points on either side. Is it always possible to do this in reverse order? That is, given a finite set of points in the plane, is it always possible to draw a line that divides it neatly in two? (If there are an odd number of points, assume the line must intercept precisely one of them.)

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Yes, it’s always possible. Join every pair of points with a line segment. Now, to the left of the field, draw a line that’s parallel to none of these segments. If we move the line rightward through the field, keeping its slope constant, then it will encounter one point at a time, so we can stop it precisely midway through the field (or anywhere else we like).

February 26, 2012January 31, 2013 | Puzzles

Black and White

sam loyd chess problem

By Sam Loyd. White to mate in two moves.

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1. Be8! and Black can choose among four mates.

February 24, 2012January 31, 2013 | Puzzles

Jackson Strive

You and I spot a $20 bill on the street. To divide it, we agree to an auction: Each of us will write down a bid, and the high bidder will keep the $20 but pay the amount of his own bid to the other player. If we submit the same bid then we’ll split the $20. What should you bid?

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If you bid too low (say, $1), then in the worst case I’ll bid a penny higher (leaving you with $1.01). If you bid too high (say, $19), then in the worst case you’ll win the auction but have to give me most of your winnings. Bidding $10 guarantees you at least $10 profit: If I bid higher than I’ll have to pay you my bid, and if I bid lower then you’ll win the $20 and have to pay me only $10. But bidding $9.99 also guarantees you $10 if I bid higher — and it nets you an extra penny if I bid lower. That’s your best bet.

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February 22, 2012January 31, 2013 | Puzzles

Green Ties

The Martian Census Bureau compiled the marital history of every male and female Martian, living and dead:

Never married: 6,823,041
Married once: 7,354,016
Married twice: 1,600,897
Married three times: 171,013
Married four times: 2,682

What’s wrong with these figures?

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The total number of Martians married an odd number of times must be even (it takes two to get married, even on Mars). But 7,354,016 + 171,013 gives 7,525,029, an odd number. Contributed by Sidney Kravitz, Dover, N.J., to the Journal of Recreational Mathematics 2:4, October 1969.

February 21, 2012January 31, 2013 | Puzzles

Fields of Gold

fields of gold puzzle

Which of the yellow areas is larger?

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They’re equal. The long diagonal bisects the large rectangle as well as each of the smaller white rectangles. If the two halves of the large rectangle are equal, and if the smaller white rectangles are divided evenly between them, then the area that remains in each half (yellow) must be equal as well.

February 18, 2012January 31, 2013 | Puzzles

Black and White

From Stratagems of Chess, 1817. White to mate in two moves.

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1. Qc5+ dxc5 2. Rd8#

February 17, 2012January 31, 2013 | Puzzles

Picture This

We must assemble a jigsaw puzzle that contains 100 pieces. If a “move” consists of connecting two sections (even a section containing a single piece), what’s the minimum number of moves we must make to complete the puzzle?

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Each move reduces the total number of sections by 1. Since we start with 100 sections and end with 1, it will take 99 moves to assemble the puzzle, no matter how we proceed. See Breaking Bad.

February 14, 2012January 31, 2013 | Puzzles

Defense Posture

You walk into Arlington, Virginia. As the Pentagon comes into view, you’ll be able to see either two or three sides of the building. Which is more likely?

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They’re equally likely. Imagine someone approaching the building from the opposite side. If you can see two sides, she can see three, and vice versa. Because these are the only two possibilities, the probability of each must be 1/2. (Strictly speaking, the two probabilities are perfectly equal only if you’re approaching an abstract pentagonal building from an infinite distance. But you get the point.)

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February 11, 2012January 31, 2013 | Puzzles