Henry Dudeney in *Strand*, June 1911: “It would be difficult to find a prettier little chess problem in three moves, produced from such limited material as a rook and a pawn, than the one given this month, by Dr. S. Gold. The novice will probably find the task of discovering the key move quite perplexing. White plays and checkmates in three moves.”

# Puzzles

# The Switchback Puzzle

“Some years ago there was a craze for rolling pellet puzzles,” wrote Henry Dudeney in 1909, “though they are really more trials of patience than puzzles.”

One exception was this undulated glass tube, which contained three shots or pellets. The task was to get them into the three depressions at A, B, and C, which are unfortunately positioned at high points in the tube.

This “could be solved by a puzzle trick which I was surprised to notice how few people discovered,” Dudeney wrote. What was it?

# Made to Order

Arrange the digits 0-9 into a 10-digit number such that the leftmost *n* digits comprise a number divisible by *n*. For example, if the number is ABCDEFGHIJ, the number ABC must be divisible by 3, ABCDE must be divisible by 5, and so on.

# Black and White

From Henry Dudeney’s Perplexities column, *Strand*, March 1911:

“Here is a pretty little chess puzzle, made some years ago by Mr. F. S. Ensor. White has to checkmate the Black king without ever moving a queen off the bottom row, on which they at present stand. It is not difficult. As the White king is not needed in this puzzle, His Majesty’s attendance is dispensed with. His three wives can dispose of the enemy without assistance — in seven moves.”

# “A Match Puzzle”

This puzzle, by T.E. Maw of the Luton Public Library, appeared in the *Strand* in April 1911:

“Take fifteen matches and place them as shown in the diagram, then take away three, change the position of one, and the result will be the word showing what matches are made of.”

# Exam Week

A problem submitted by the United States and shortlisted for the 16th International Mathematical Olympiad, Erfurt-Berlin, July 1974:

Alice, Betty, and Carol took the same series of examinations. There was one grade of A, one grade of B, and one grade of C for each examination, where A, B, C are different positive integers. The final test scores were

Alice: 20

Betty: 10

Carol: 9If Betty placed first in the arithmetic examination, who placed second in the spelling examination?

# To Town

A problem from the 1949 problems drive of the Archimedeans, the mathematical society of Cambridge University:

A farmer lives in a cottage 4/17 of a mile from a main road. There is a lane leading from his farm to the nearest point Q on the road. The road is straight running north and south, and there is a village two miles south of Q at which he keeps a bicycle. He wishes to go to a town on the road four miles north of Q. He can walk across the fields surrounding the roads at 1 1/2 miles per hour, but along the roads he can walk at 3 1/2 miles per hour. He can cycle at 14 miles per hour. Should he collect his bicycle in order to get to the town from his farm as quickly as possible?

# $50 for All

A puzzle by Ben H., a systems engineer at the National Security Agency, from the agency’s August 2016 Puzzle Periodical:

At a work picnic, Todd announces a challenge to his coworkers. Bruce and Ava are selected to play first. Todd places $100 on a table and explains the game. Bruce and Ava will each draw a random card from a standard 52-card deck. Each will hold that card to his/her forehead for the other person to see, but neither can see his/her own card. The players may not communicate in any way. Bruce and Ava will each write down a guess for the color of his/her own card, i.e. red or black. If either one of them guesses correctly, they both win $50. If they are both incorrect, they lose. He gives Bruce and Ava five minutes to devise a strategy beforehand by which they can guarantee that they each walk away with the $50.

Bruce and Ava complete their game and Todd announces the second level of the game. He places $200 on the table. He tells four of his coworkers — Emily, Charles, Doug and Fran — that they will play the same game, except this time guessing the suit of their own card, i.e. clubs, hearts, diamonds or spades. Again, Todd has the four players draw cards and place them on their foreheads so that each player can see the other three players’ cards, but not his/her own. Each player writes down a guess for the suit of his/her own card. If at least one of them guesses correctly, they each win $50. There is no communication while the game is in progress, but they have five minutes to devise a strategy beforehand by which they can be guaranteed to walk away with $50 each.

For each level of play — 2 players or 4 players — how can the players ensure that someone in the group always guesses correctly?

# Paint Job

Given a standard chessboard, you can choose any rank or file and repaint each of its squares to the opposite color (white squares turn black, and black squares turn white). By doing this repeatedly, is it possible to produce a board with 63 white squares and one black square?

# Black and White

A tidy problem by G.A.W. Cumming. White to mate in two moves.