Saw 6

We want to cut a 3-inch cube into 27 1-inch cubes. We can do this by making six cuts, as shown. Can we accomplish the task with fewer cuts by rearranging the pieces between cuts?

	Select Click for Answer
No. Imagine the 1-inch block at the heart of the 3-inch block. It has six faces, and all six must be freed. This requires six cuts.

March 4, 2012January 31, 2013 | Puzzles

Black and White

By T. Herlin. White to mate in two moves.

	Select Click for Answer
1. Qh1 and a rook must abandon the king’s protection.

March 2, 2012January 31, 2013 | Puzzles

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?

	Select Click for Answer
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?

	Select Click for Answer
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.

Click for Answer

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.)

	Select Click for Answer
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.

	Select Click for Answer
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?

	Select Click for Answer
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.

Click for Answer

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?

	Select Click for Answer
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?

	Select Click for Answer
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.

	Select Click for Answer
1. Qc5+ dxc5 2. Rd8#

February 17, 2012January 31, 2013 | Puzzles