Winners and Losers

A puzzle by the Hungarian-Canadian mathematician George Grätzer:

I’m writing an article about a round-robin tennis tournament, in which each player plays each other player once. I decide to pick one player and ask her which players she defeated (in tennis there are no ties). Then I’ll ask each of those players which players they’ve defeated. Is it possible to pick a player so that everyone in the tournament is mentioned in my article?

Click for Answer

Dead Reckoning

buchanan chess puzzle

A.G. Buchanan posed this curious puzzle in The Problemist in July 2001. Who moved last? The question seems absurd. If each side has only a bare king, how can we know which made the last move?

The answer turns on Rule 5.2b in the Laws of Chess:

The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position.’ This immediately ends the game, provided that the move producing the position was legal.

In the position above, suppose it was Black who moved last. He cannot simply have moved his king to the corner from a7 or b8, because in that earlier position Rule 5.2b would already have applied: The game would have ended in a draw at that point, and Black would have had no opportunity to move his king to a8. Similarly, Black cannot have captured a knight or a bishop on a8, because neither of those pieces (alone with a king) is sufficient to give checkmate, and again the game would have ended before the diagrammed position could be reached.

Black might have captured a rook or a queen on a8. But consider that case: Suppose there was a queen on a8, and the black king was in check on a7 or b8. In that case the capture was forced — Black had no other legal move. And hence even before the capture took place it would have been correct to say that “neither player can checkmate” — the capture was ordained and no possible mate lay in the future. And so the game would have ended at that point, and again we could never have reached the diagrammed position.

Hence Black has no possible legal last move, and the answer to the puzzle is that White moved last, capturing a black piece on c6. Because this capture wasn’t forced, Rule 5.2b is not invoked.

This is a technicality, but it’s an important one. In 2015 the World Federation of Chess Composition voted that the “dead position” rule applies only to retrograde (backward-looking) problems like the one above. More details are here.

The Seven Coins Puzzle

seven coins puzzle

We want to place a coin at each vertex of this figure but one. A coin is placed by moving it along a free line and putting it down at the end of that line. A line is called free if there’s no coin at either of its numbered endpoints. So, for example, we might put a coin on 1 by moving it from 4 to 1 and leaving it there. Then we could put a coin on 2 by moving along 5-2, then on 3 by moving along 6-3, on 4 by moving along 7-4, and on 5 by moving along 8-5. But then we’re stuck — there are no more free lines, and we’ve placed only five coins. How can we place all seven?

Click for Answer


Rotenberg kriegspiel problem

Kriegspiel is a variant of chess in which neither player can see the other’s pieces. The two players sit at separate boards, White with the white pieces and Black with the black, and a referee facilitates the game. When a player attempts a move, the referee declares whether it’s legal or illegal. If it’s legal then it stands; if it’s not, the player retracts it and tries again.

This makes for some interesting chess problems. In this example, by Jacques Rotenberg, White knows that there’s a black bishop on a dark square, but he doesn’t know where it is. How can he mate Black in 8 moves?

This is tricky, because if White captures the bishop by accident, the position is stalemate. Accordingly White must avoid bishop or knight moves to begin with. The answer is to try 1. Rg2. If the referee declares that this is illegal, that means that the black bishop is somewhere on the second rank and it’s safe for White to play 1. Nf2, giving mate immediately.

If the referee declares that 1. Rg2 is legal, then the move is made, Black moves his invisible bishop (his king and pawn have no legal moves), and it’s White’s turn again.

Now White announces 2. Rg8. If the referee says that this is illegal, then the black bishop is on the g-file, and White can safely play 2. Be5. Now if Black captures the bishop, then 3. Nf2 is mate; on any other Black move, 3. Nf2+ followed (if necessary) by 4. Rxh2+ is mate.

If 2. Rg8 is legal, then White plays it, Black again inscrutably moves his bishop, and now White plays 3. Rh8. (There’s no danger that he’ll capture the black bishop inadvertently on h8, because it cannot have been on g7 on the previous turn.)

Black moves his invisible bishop again and now White plays 4. Rh5 followed by 5. Rb5 (if that’s not possible then 5. Rh3 and 6. Be5), 6. Rb1, 7. Nf2+ Bxf2 and 8. Kxf2#. White wins in eight moves at most. In order to travel safely from a2 to b1, the white rook must pass through h8!

Podcast Episode 221: The Mystery Man of Essex County

In 1882, a mysterious man using a false name married and murdered a well-to-do widow in Essex County, New York. While awaiting the gallows he composed poems, an autobiography, and six enigmatic cryptograms that have never been solved. In this week’s episode of the Futility Closet podcast we’ll examine the strange case of Henry Debosnys, whose true identity remains a mystery.

We’ll also consider children’s food choices and puzzle over a surprising footrace.

See full show notes …

Cache and Carry

USC mathematician Solomon W. Golomb offered this problem in the Pi Mu Epsilon Journal, Fall 1971 (page 241):

Ted: I have two numbers x and y, where x + y = z. The sum of the digits of x is 43 and the sum of the digits of y is 68. Can you tell me the sum of the digits of z?

Fred: I need more information. When you added x and y how many times did you have to carry?

Ted: Let’s see. … It was five times.

Fred: Then the sum of the digits of z is 66.

Ted: That’s right! How did you know?

Click for Answer

Common Sense

A quickie from Raymond Smullyan: On the Island of Knights and Knaves, knights always tell the truth and knaves always lie. Every inhabitant is either a knight or a knave. One day a visiting anthropologist comes across a native and recalls that his name is either Paul or Saul, but he can’t remember which. He asks him his name, and the native replies “Saul.”

From this we can’t know whether the native is a knight or a knave, but we can tell with high probability. How?

Click for Answer