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?

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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?

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The Red Ball

An urn contains k black balls and one red ball. Peter and Paula are going to take turns drawing balls from the urn (without replacement), and whoever draws the red ball wins. Peter offers Paula the option to draw first. Should she take it? There seem to be arguments either way. If she draws first she might get the red ball straightaway, and it seems a shame to give up that opportunity. On the other hand, if she doesn’t succeed immediately then she’s only increased Peter’s chances of drawing the red ball himself. What should she do?

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Line Limit

You own a goat and a meadow. The meadow is in the shape of an equilateral triangle each side of which is 100 meters long. The goat is tied to a post at one corner of the meadow. How long should you make the tether in order to give the goat access to exactly half the meadow?

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Black and White

I just ran across this in Benjamin Glover Laws’ The Two-Move Chess Problem, from 1890. It’s by G. Chocholous. White is to mate in two moves.

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Cube Route

Created by Franz Armbruster in 1967, “Instant Insanity” was the Rubik’s Cube of its day, a simple configuration task with a dismaying number of combinations. You’re given four cubes whose faces are colored red, blue, green, and yellow:
Image: Wikimedia Commons

The task is to arrange them into a stack so that each of the four colors appears on each side of the stack. This is difficult to achieve by trial and error, as the cubes can be arranged in 41,472 ways, and only 8 of these give a valid solution.

One approach is to use graph theory — draw points of the four face colors and connect them to show which pairs of colors fall on opposite faces of each cube:
Image: Wikimedia Commons

Then, using certain criteria (explained here), we can derive two directed subgraphs that describe the solution:
Image: Wikimedia Commons

The first graph shows which colors appear on the front and back of each cube, the second which colors appear on the left and right. Each arrow represents one of the four cubes and the position of each of the two colors it indicates. So, for example, the black arrow at the top of the first graph indicates that the first cube will have yellow on the front face and blue on the rear.

This solution isn’t unique, of course — once you’ve compiled a winning stack you can rotate it or rearrange the order of the cubes without affecting its validity. B.L. Schwartz gives an alternative method, through inspection of a table, as well as tips for solving by trial and error using physical cubes, in “An Improved Solution to ‘Instant Insanity,'” Mathematics Magazine 43:1 (January 1970), 20-23.