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Puzzles

Ant in a Room

Puzzle

An ant crawls from one corner of a room to the diametrically opposite corner along the shortest possible path. If the dimensions of the room are 3 x 4 x 5, what distance does the ant cover?

Source

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

Solution

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

Followup

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.

Previous Puzzle: Ant Collisions

Peter and Cynthia stand at each end of a straight line segment. Peter sends 50 ants towards Cynthia, one after another. Cynthia sends 20 ants towards Peter. All ants travel along the straight line segment. Whenever two ants collide, they simply bounce back and start traveling in the opposite direction. How many ants reach Peter and how many reach Cynthia? How many ant collisions take place?

Next Puzzle: Coin Toss Guess

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?