Six Choices

On a multiple-choice test, one of the questions is illegible, but the choice of answers is listed clearly below. What’s the right answer?

(a) All of the below.
(b) None of the below.
(c) All of the above.
(d) One of the above.
(e) None of the above.
(f) None of the above.

	Select Click for Answer
(a) implies that both (b) and (f) are right, which is a contradiction, so (a) can’t be right. And (c) claims that (a) is right, so (c) can’t be right either. We’ve eliminated (a) and (c). That means that if (b) is a correct statement then (d) is accurate. But (b) contradicts (d), so (b) must be false. And since (a), (b), and (c) are all false, then (d) is false as well. Now we’re down to (e) and (f). (f) can’t be true, because it implies simultaneously that statements (a) through (d) are false and that (e), which agrees with this, is also false, another contradiction. That leaves (e), whose truth is confirmed by the falsity of statements (a) through (d).

Click for Answer

March 24, 2012February 1, 2013 | Puzzles

Black and White

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

	Select Click for Answer
1. Rd8+ and the queen will mate.

March 23, 2012February 1, 2013 | Puzzles

Paint Scheme

paint scheme puzzle

How many colors are necessary to paint the squares of a chessboard so that no bishop can move between two squares of the same color?

	Select Click for Answer
Eight. Every bishop move involves a change of rank, so painting each rank a different color will do the trick. Using fewer than eight colors won’t do, as each long diagonal covers eight ranks.

March 19, 2012February 1, 2013 | Puzzles

Black and White

By J. Berger. White to mate in two moves.

	Select Click for Answer
1. After the ridiculous-looking Qc6 Black must succumb to a check down the b-file or a mate at close quarters by the queen.

March 16, 2012February 1, 2013 | Puzzles

Fowl Play

From a 1947 competitive examination for high school seniors conducted by Stanford’s math department:

My grandfather’s papers included an old invoice:

72 turkeys $-67.9-

The first and last digits are illegible. What are the missing digits, and what was the price of one turkey?

	Select Click for Answer
If -679- is divisible by 72, then it’s also divisible by 8 and by 9. If -679- is divisible by 8, then 79- is also divisible by 8 (we can ignore the two leftmost digits because any multiple of 1000 is divisible by 8). Of all the possibilities for 79-, only 792 is evenly divisible by 8, so the last digit must be 2. If -6792 is divisible by 9, then the sum of its digits must be a multiple of 9. That means the first digit must be 3. If the total invoice was for $367.92, then one turkey cost $5.11. (Via George Polya and Jeremy Kilpatrick, The Stanford Mathematics Problem Book, 2013. Thanks, Sid.)

Click for Answer

March 14, 2012November 14, 2018 | Puzzles

Rolling

I drove this car 20,000 miles and used five tires equally in accumulating the mileage. How many miles’ wear did each tire sustain?

	Select Click for Answer
The car runs on four tires, so it accumulated 80,000 tire-miles in total. If the five tires were worn equally, then each was used for 16,000 miles.

March 12, 2012January 31, 2013 | Puzzles

Black and White

By J.H. Graham. White to mate in two moves.

	Select Click for Answer
1. Qb4! Black must capture a knight, and the queen mates from e1 or e7.

March 9, 2012January 31, 2013 | Puzzles

Open and Shut

A warden oversees an empty prison with 100 cells, all closed. Bored one day, he walks through the prison and opens every cell. Then he walks through it again and closes the even-numbered cells. On the third trip he stops at every third cell and closes the door if it’s open or opens it if it’s closed. And so on: On the nth trip he stops at every nth cell, closing an open door or opening a closed one. At the end of the 100th trip, which doors are open?

	Select Click for Answer
Consider some representative cells. The warden stops at cell 7 twice, on the first and seventh trips. The divisors of 7 are 1 and 7. He stops at cell 10 four times, on the first, second, fifth, and tenth trips. The divisors of 10 are 1, 2, 5, and 10. Each door receives a quantity of visits equal to its number’s divisors. All integers except perfect squares have an even number of divisors. So after 100 trips, only the doors whose numbers are perfect squares will remain open: cells 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

Click for Answer

March 6, 2012January 31, 2013 | Puzzles

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