Installment Plan

A traveler wants to stay at an inn for seven days. He has no money, but he has a gold chain with seven links. The innkeeper agrees to accept this in payment for the week’s stay, but the traveler is reluctant to part with all seven links at once. He prepares to cut the chain into seven pieces.

The innkeeper stops him. If the traveler is willing occasionally to accept change in the form of links previously paid, then they can work out a plan that minimizes damage to the chain and yet permits the traveler to pay only what he owes on each successive day. How many links must they cut?

	Select Click for Answer
They cut a single link, the third one, dividing the chain into lengths of 1, 2, and 4. Now: On day 1 the traveler gives the innkeeper 1 link. On day 2 the traveler gives the innkeeper 2 links and accepts 1 in return. On day 3 the traveler gives the innkeeper 1 link. On day 4 the traveler gives the innkeeper 4 links and accepts 3 in return. On day 5 the traveler gives the innkeeper 1 link. On day 6 the traveler gives the innkeeper 2 links and accepts 1 in return. On day 7 the traveler gives the innkeeper 1 link. (In effect they’re counting in base 2.)

Click for Answer

October 28, 2011January 29, 2013 | Puzzles

Squeeze Play

A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make?

	Select Click for Answer
Surprisingly (to me), the arc will be more than 44 feet high. See Waistline.

October 25, 2011January 29, 2013 | Puzzles · Science & Math

Black and White

black and white -- hassberg, new york post

By Eric M. Hassberg, New York Post, April 21, 1945. White to mate in two moves.

	Select Click for Answer
1. Qc2! and White will mate from h7 or h2.

October 23, 2011January 29, 2013 | Puzzles

Hands Up

There are 13 ways to draw four of a kind and 40 possible straight flushes.

Why then does a straight flush beat four of a kind?

	Select Click for Answer
The four-of-a-kind hand contains a fifth card, which can take any of 48 values. Hence there are 624 poker hands that contain four of a kind, and only 40 straight flushes.

October 22, 2011January 29, 2013 | Puzzles

The Interloper

Consider the classic story of Henry, a backwoods villager watching a theatrical performance, who leaps to the stage to save the heroine from the clutches of the villain and a horrible death. Of course Henry is mistaken if he thinks he can save the actress. She is not in danger; there is nothing to save her from. But the character she portrays is in danger and does need saving. Can Henry help her, despite the fact that he doesn’t live in her world?

— Kendall L. Walton, “How Remote Are Fictional Worlds From the Real World?”, Journal of Aesthetics and Art Criticism, 1978

October 20, 2011October 19, 2011 | Puzzles

Boxing Day

Death row is overcrowded, so the warden proposes a radical solution. He places 100 boxes in a sealed room. Each contains a slip of paper bearing the name of one of the 100 prisoners on the row.

Each prisoner will enter the room by one door, open 50 boxes, and exit by another door. Unless every prisoner can discover his own name, all 100 will be executed.

The prisoners will be supervised by a guard. They cannot communicate with one another, and they must leave the room as they found it, but the group can prepare a plan in advance and post it on the wall of the room.

If they proceed at random, their chance of succeeding is 1/2¹⁰⁰, or about 0.00000000000000000000000000000008. What should they do?

	Select Click for Answer
The written plan assigns one box to each prisoner. When he enters the room, he opens his assigned box, reads the name on the slip, and opens the box assigned to that prisoner. He continues in this way until he’s opened 50 boxes or found his own name. Remarkably, this strategy raises the chance of success to 31.18 percent. Peter Winkler discusses the idea here; the puzzle originated with Danish computer scientist Peter Bro Miltersen. (Thanks, Steve.)

Click for Answer

October 18, 2011January 29, 2013 | Puzzles

Accordion Commute

A puzzle by David Wells:

Every day I take the subway from Startville to Endville. Today I arrived at the Startville station to find that my train was just departing. I caught the next train to Endville, where I left the station at exactly the same time as if I had caught the first train. How did I manage this? The two trains traveled at the same speed, and I myself did not have to rush to make up the lost time.

	Select Click for Answer
Startville is north of Endville. Normally I arrive at the north end of the Startville station, enter the last car of my regular train, and ride it to Endville, where I leave the train and walk the length of the platform to exit at the south end of the station. Today, when I saw that I had missed my regular train, I walked the length of the Startville platform and arrived at the south end just as the next train arrived. I boarded the first car, rode it to Endville, and debarked at the south end of the Endville station, next to the exit. My normal commute involves walking the length of a subway platform; today I simply did it in Startville rather than Endville.

Click for Answer

October 17, 2011January 29, 2013 | Puzzles

Black and White

black and white

By Auguste D’Orville. White to mate in two moves.

	Select Click for Answer
1. Nd2!, followed by 2. c4#. From Problèmes d’Echecs, composés et dédiés aux amateurs de ce jeu, 1842.

October 5, 2011January 29, 2013 | Puzzles

Black and White

calvi chess problem

By Calvi. White to mate in two moves.

	Select Click for Answer
1. Qg2! and Black cannot stop both 2. Qd2# and 2. Nb7#. From Le Palamède, via Charles Tomlinson, Amusements in Chess, 1845.

September 27, 2011January 29, 2013 | Puzzles

Sudden Death

sudden death chess problem

“Paul Morphy IV” submitted this puzzle to the Journal of Recreational Mathematics in 1974. White might capture Black’s queen at this point, but instead can choose to mate on the move. How?

	Select Click for Answer
It’s a trick. The square in the lower right is dark when it should be light, so we must rotate the board 90 degrees. If we turn it to the right then the black king is in check on White’s move, which is illegal. So turn it to the left: Now White can mate by promoting the d7 pawn to a knight. (“Why Prolong It?”, JRM 7:1, p. 62)

September 20, 2011January 29, 2013 | Puzzles