Puzzle | Nineteen Pirates and the King

Nineteen pirates (one of them is their leader) were caught by the king one day. But the king heard many stories of spontaneous wit of the pirates’ leader and great and clever escapes that he led in the past and so offered a challenge to the pirates.
The challenge involves the pirates numbered 1 to 19 to be seated in chairs numbered 1 to 19 around a table but no pirate occupying a matching chair (a matching chair is one that has a number same as the pirate number who sits in it). But they all can rotate in any direction any number of times so that at least some pirates get to sit in matching chairs. The initial position of the pirates is as per the wish of the king (but no pirate will be seated in a matching chair).

King then asked the pirates’ leader that if they take this challenge and after some rotations, if at least two of the pirates are seated in matching chairs, pirates will win and all the pirates will be left free and the pirates’ leader becomes a “privateer” (a legalized pirate protected by king).
If they take this challenge and after any number of rotations if at least two pirates are not seated in matching chairs, pirates will lose badly and nine randomly chosen pirates will be punished and other pirates will be left free.

If they don’t take this challenge, pirates will loose and four randomly chosen pirates will be punished and other pirates will be left free.

The pirates’ leader accepted to take the challenge.

Question:

Will the Pirates win in all cases (all cases here mean all the initial seating configurations which have no pirate sitting in matching chair)?

Answer:

Yes.

Solution:

We can observe that after all the pirates are seated in the initial configuration, there can only be 18 distinct rotations (all clockwise or all anti-clockwise but the direction of rotation does see furthers we see further) possible after which they again come back to the initial configuration.
So, in all these 18 rotations all the below examinations happened (in fact, simultaneously) :
(1) Pirate-1 is examined at exactly 18 other chairs and one of them must match because initially, he was not in matching the chair.
(2) Pirate-2 is examined at exactly 18 other chairs and one of them must match because initially, he was not in matching the chair.
(3) Pirate-3 is examined at exactly 18 other chairs and one of them must match because initially, he was not in matching the chair.
………………………………..and so on for all other pirates too …………………………..
(18) Pirate-18 is examined at exactly 18 other chairs and one of them must match because initially, he was not in matching the chair.
(19) Pirate-19 is examined at exactly 18 other chairs and one of them must match because initially, he was not in matching the chair.

So, 19 pirates got matching chairs during 18 rotations. By pigeonhole principle, there is a rotation that caused at least two pirates, to get matching chairs.

In other words, let a pigeon be denoted by a single pirate sitting in matching chair and a pigeonhole be denoted by a rotation. There are 19 pigeons and 18 pigeon holes. So, one of the pigeonholes must have at least two pigeons.

Note that we did not bother about the direction of rotation because by rotating in any one direction consistently, a pirate is examined against all the other 18 chairs for his matching chair.



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