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#1 unreality

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Posted 18 February 2008 - 06:02 PM

yeah! I made another one! :D lol

You've played in three games of zarball!

You've played in five games of zarball!

But are you game for the Royal Zarball Tournament?

This is how the tournament works:

First Round

A vs B = I
C vs D = J
E vs F = K
G vs H = L

Second Round

I vs J = M
K vs L = N

Final Round

M vs N = WINNER!!!!!

********

Here are the competitors and your chances of beating each of them:

You, the Prisoner
Prince - 1/1 chance as usual. You can always beat the Prince, every time
Queen- 1/2 chance as usual. You can beat the Queen half of the time
King- 1/4 chance as usual. You have a quarter chance of beating the King
Duke- 1/3
Earl- 2/3
Jester- 0. The Jester will beat you every time- I mean, all he does in his spare time is juggle! But don't worry- the Jester's only weakness is the Peasant, who beats him every time
Peasant- 1/2, except when playing the Jester. The Peasant beats the Jester every time

Except for the special cases of the Jester, Peasant and Princess, the chances stay the same for beating you when they play each other (however they must be made relative to each other).

For example, if you have a 1/3 chance of beating the Duke and a 2/3 chance of beating the Earl, it follows that the Duke is twice as good as the Earl, so if they played against one another, the Earl would have a 2/3 chance of winning, and the Duke would have a 1/3 chance of winning. However this isn't true.

Spoiler for HINT: why this isn't true. Only read if you can't figure it out yourself


Okay, so down to the actual question:

You failed to escape your imprisonment the first two chances... this is your last chance, says the King. He is allowing you to arrange the starting bracket of the Royal Zarball Tournament. Remember the key in this post- the 8 starting positions are the letters A through H.

How should you arrange the eight competitors on the bracket to give you the maximum chance of winning the Royal Zarball Tournament and go free? What is that chance? Is there more than one solution?
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#2 bonanova

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Posted 18 February 2008 - 06:22 PM

King Queen Prince Princess = 4
Duke, Earl, Jester, Peasant = 4
Then there's you. That's 9 total.
But there are only 8 slots.
Help. :wacko:
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#3 unreality

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Posted 18 February 2008 - 07:28 PM

oops sorry lol. The Princess was something I was thinking about for a future problem... and I put it in there without thinking ;D wow. hehe. Nice catch, bonanova.
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#4 PolishNorbi

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Posted 18 February 2008 - 08:32 PM

Spoiler for My Solution

Edited by PolishNorbi, 18 February 2008 - 08:38 PM.

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#5 unreality

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Posted 18 February 2008 - 09:03 PM

There's enough time for everybody to regain their strength between matches. The chances given remain the same throughout the entire Tournament :D
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#6 unreality

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Posted 18 February 2008 - 09:33 PM

My own solution:

Spoiler for unreality's answer


As for multiple solutions:

Spoiler for yeah

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#7 unreality

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Posted 18 February 2008 - 11:14 PM

Any get a better chance than 7/54?
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#8 Jkyle1980

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Posted 19 February 2008 - 01:08 AM

Any get a better chance than 7/54?


Jester
Peasant

King
Duke

Queen
Earl

Prince
Me

With this, the worst case scenario is playing the prince, queen, and king (odds of winning all three is 1/8, 12.5%). I got stuck trying to figure the odds of who would win between the King and Duke. But the worst case scenario odds for the above solutions is 6/54, 11.1%. If I get the luck of the peasant making it to the finals, my odds of winning it all are 1/4, 25%. If the Duke makes it to the finals, my odds of winning are 1/6, 16.7%. Like I said, I'm not sure of the math between the king and duke so...ah, hell, give me a second...
...
Yeah, still don't have it. But I think my bracket gives you better odds. I'd be interested to see the math if anyone is better with fractions than I am.
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#9 bonanova

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Posted 19 February 2008 - 02:59 AM

This puzzle is really a nice challenge. Kudos!! B))

Jester's strength against anyone other than Peasant and Me is not given,
making it mandatory to match them in the first round.
If more info is given about Jester, that could change my best result.

For now, it's 0.191
That's slightly better than the 7/54 = 0.1296 result.

My analysis - in four steps ...
Spoiler for 1 - Get individual playing strengths
Spoiler for 2 -Head-to-head winning probabilities
Spoiler for 3- Compute probabilities from brackets
Spoiler for 4 - Results from different strategies

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#10 bonanova

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Posted 19 February 2008 - 03:04 AM

UR,

I see you pairing the King against Jester with Jester winning.
Here's what you say about Jester:

The Jester will beat you every time- I mean, all he does in his spare time is juggle!
But don't worry- the Jester's only weakness is the Peasant, who beats him every time


All you say explicitly is that [a] Jester beats me. [b] Peasant beats Jester.
Where do you get that Jester beats King? "only weakness" doesn't seem that specific.
You might have said Jester beats everybody - except Peasant - if that's what he does.

Can you clarify? I can do a new calculation with Jester against King then.

Thanks for a nice puzzle...!
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell




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