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You are a prisoner to a land ruled by a King and Queen, both of whom are pretty nice, luckily for you. The King enjoys riddles, math challenges, and physical games as well. Often he comes and chats with you in your cell. You have only been there two weeks when he says he has an idea. He will let you go free if you can win two games in a row of zarball. Zarball is a 1-on-1 game of agility, strength, speed and accuracy, played throughout the land. You think you're pretty good at it. You know that the King is one of the best, though, and the Queen is also pretty good, though nowhere near as good as the King.

The King says: "If you can win two games in a row out of three games, you shall go free." He scratched his goti and looked at you with an interested face. "The three games will be alternating of me and my wife. You can play me first, the Queen second, and me last. Or you can play the Queen first and last, and me second, in the middle. KQK or QKQ. Either way is fine."

What do you do?

8 answers to this question

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You are a prisoner to a land ruled by a King and Queen, both of whom are pretty nice, luckily for you. The King enjoys riddles, math challenges, and physical games as well. Often he comes and chats with you in your cell. You have only been there two weeks when he says he has an idea. He will let you go free if you can win two games in a row of zarball. Zarball is a 1-on-1 game of agility, strength, speed and accuracy, played throughout the land. You think you're pretty good at it. You know that the King is one of the best, though, and the Queen is also pretty good, though nowhere near as good as the King.

The King says: "If you can win two games in a row out of three games, you shall go free." He scratched his goti and looked at you with an interested face. "The three games will be alternating of me and my wife. You can play me first, the Queen second, and me last. Or you can play the Queen first and last, and me second, in the middle. KQK or QKQ. Either way is fine."

What do you do?

First glance, I would say QKQ but..

Assuming, I can beat the queen, I would play KQK. The reasoning behind this is you win against the queen and you have two opportunties against the king.

While a single loss against the King (more likely to happen than two losses) means you are automatically done. No more chances, no other way of doing it.

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yep good job! ;D

Since you're going for 2 in a row, not 2/3, you need the middle spot to win. So you want the middle spot to be the Queen, because you have a higher chance of beating her than beating the King. Then all you have to do is beat the King just once- with two opportunities to face him, you are more likely to win.

So go with KQK

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The problem can be solved by logic/intuition/whatever, which is what I did (and PolishNorbi), though you can also solve it with pure probability ;D

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Play the King twice!

Why?

Intuitively, regardless of the order, KQK or QKQ, you have to beat both K and Q.

But since the wins must be consecutive, you absolutely must win the 2nd game.

So make the 2nd game against the weaker player: Q.

But let's check that out.

Suppose k and q are the probabilities of beating the King and the Queen.

Of the 8 possible outcomes, LLL, WLL, LWL, LLW, WWL, WLW, LWW, WWW, only three get you out of jail.

Compute their probabilities.

p[WWW] = kqk
p[WWL] = kq(1-k)
p[LWW] = (1-k)kq
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p[being freed] = kq(2-k)

For QKQ [switch k and q]

p[being freed] = kq(2-q)
`For KQK:`

So if q>k [Queen is weaker player] then (2-k) > (2-q), and KQK is the preferred order of play.

Example if k=.1 and q=.9 the out-of-jail probabilities are

0.171 for KQK and 0.099 for QKQ.

But change the rules to 2 wins out of 3 and things change drastically:

0.172 for KQK and a whopping 0.828 for QKQ - increasing by 0.729 for beating the Queen twice.

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if your the prisoner im sore you wish you had to win two out of three but since its two in a row KQK

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Everyone got it right ;D and bonanova nailed the Probability Proof of it. Good job!

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I'm posting a sequel now

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You are a prisoner to a land ruled by a King and Queen, both of whom are pretty nice, luckily for you. The King enjoys riddles, math challenges, and physical games as well. Often he comes and chats with you in your cell. You have only been there two weeks when he says he has an idea. He will let you go free if you can win two games in a row of zarball. Zarball is a 1-on-1 game of agility, strength, speed and accuracy, played throughout the land. You think you're pretty good at it. You know that the King is one of the best, though, and the Queen is also pretty good, though nowhere near as good as the King.

The King says: "If you can win two games in a row out of three games, you shall go free." He scratched his goti and looked at you with an interested face. "The three games will be alternating of me and my wife. You can play me first, the Queen second, and me last. Or you can play the Queen first and last, and me second, in the middle. KQK or QKQ. Either way is fine."

What do you do?

I'd choose KQK, because then you have a higher chance of winning two in a row. If you play QKQ you have to win from the King in order to get two in a row. Now, you have a 50% chance of winning, assuming that you win from the queen. With QKQ everything is hanging on the one game against the king.

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