From Physics — I by R Resnick, D Halliday and K Krane.

If the room's dimensions are a x b x c, then the length of the shortest path is

Two curious problems:

1) Dudeney's Spider and Fly Problem (posed in an English newspaper in 1903): "In a rectangular room (a cuboid) with dimensions 30×12×12, a spider is located in the middle of one 12×12 wall one unit away from the ceiling. A fly is in the middle of the opposite wall one unit away from the floor. If the fly remains stationary, what is the shortest total distance the spider must crawl along the walls, ceiling, and floor in order to capture the fly?" Solution at Wolfram

2) Donald Knuth's Ant Problems: "With a cuboid 1x1x2, what is the furthest point on the surface from a corner, and which are the pairs of points with the greatest surface distance between them?"

Kotani's Ant Problem by Dick Hess solves both of these problems. Also described in Kotani's Ant Problem (book chapter) in Puzzler's Tribute: A Feast for the Mind edited by David Wolfe and Tom Rodgers.

Surface Distances on a Cuboid by Henry Bottomley discusses solutions to the above problems, along with many others.

Alice and Bob are playing a game. They are teammates, so they will win or lose together. Before the game starts, they can talk to each other and agree on a strategy.

When the game starts, Alice and Bob go into separate soundproof rooms — they cannot communicate with each other in any way. They each flip a coin and note whether it came up Heads or Tails. (No funny business allowed — it has to be an honest coin flip and they have to tell the truth later about how it came out.) Now Alice writes down a guess as to the result of Bob’s coin flip; and Bob likewise writes down a guess as to Alice’s flip.

If either or both of the written-down guesses turns out to be correct, then Alice and Bob both win as a team. But if both written-down guesses are wrong, then they both lose.

Can you think of a strategy Alice and Bob can use that is guaranteed to win every time?

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