Puzzles

The pirate game [Wikipedia, TED-Ed YouTube]

(Source: Anna M., August 2016)
A village is attacked on a daily basis by a dragon. Every night, the dragon returns to one of five caves situated in a row on a mountain. He never spends consecutive nights in the same cave. Instead, he always returns to a cave next to the one where he spent the previous night. The first and last caves are each next to only one other cave, while the other caves are each next to two other caves. In other words, cave 1 is only next to cave 2, while cave 2 is next to both cave 1 and cave 3, and so on.

The villagers can capture the dragon at night if they go to the cave where he's sleeping. However, they don't know in which cave he'll be spending the night, they don't know in which cave he stayed on any previous night, and they can only go to one cave each night. How many nights do they need at a minimum to guarantee capturing the dragon? And in which order should they visit the caves to capture him in the minimum number of nights?

(Source: Karissa P., August 2016)
Using each of the numbers 1, 3, 4, and 6 exactly once, write an expression that evaluates to 24 using only addition, subtraction, multiplication, and/or division (and parentheses).

(Source: Nina M., June 2016)
The mutilated chessboard problem [Wikipedia] — Suppose a standard 8x8 chessboard has two diagonally opposite corner squares removed. Is it possible to cover all of the remaining squares with dominoes of size 2x1?

(Source: Ryan V., November 2014)
If you can cover a rectangle with 25 circles of diameter x, can you cover the rectangle with 100 circles of diameter x/2?

(Source: Jürg W., August 2006)
Prison guards line up 100 prisoners, one behind the other. Each prisoner is wearing either a red or a blue hat. The n'th prisoner can see the hats of the previous n-1 prisoners, but he can't see his own hat or that of any of the 100-n prisoners behind him. Every prisoner can hear every other prisoner. The head guard then tells them that he will ask every prisoner starting with the last one in line what color hat he is wearing. If they only make at most one mistake, they will all be freed. Otherwise they will all be killed. What should they do? Note that the only thing the prisoners are allowed to do is to say either "red" or "blue"; they're not allowed to give any other signals.

(Source: Jürg W., August 2005)
Move one digit to make the following expression true:

3013 = 10

(Source: someone at PayPal, ca. May 2002)
Alice and Bob go to a small party where four other couples are in attendance. As the guests walk around meeting each other, they sometimes shake each other's hands. After a little while, Alice goes around and asks everyone (except for herself, of course, but including Bob) how many people's hands they shook, and everyone gives her a different answer (and they're telling the truth)! How many people's hands must Alice have shaken? (Nobody shakes their own hand or their partner's hand.)

(Source: Ryan V., 2002)
Two integers between 2 and 100 (inclusive) are chosen. P is given their product, and S is given their sum.

P says: "I can't figure out what the two numbers are."
S replies: "Yeah, I already knew that."
P replies: "Oh, really, well now I know what the two numbers are."
S replies: "Oh, really, well now I do too."

What are the two numbers?

(Source: Kevin H., June 2001)

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