[unreality]

yeah! I made another one! 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.

Look at the Queen and the King. If you can beat the Queen half of the time, it means she beats you the other half. You are of equal skill. Meaning the Queen has a 1/4 chance of beating the King, just like you.

Take the Duke and Earl example. The Duke has a 2/3 chance of beating you cuz you have a 1/3 chance of beating him. The Earl has a 1/3 chance of beating you. So you are twice as good as the Earl and half as good as the Duke. The Duke is 4x better than the Earl, not 2x. Thus the Duke would have 4/5 chance of beating the Earl.

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?

[bonanova]

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.

[unreality]

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.

[Guest]

My logical solution, without using any huge math.

Knowing that if Jester doesn't play Peasant, I lose, I want to guarentee myself a Jester Peasant Semi in the other bracket. This is so the Jester can take out the king, and I don't have to worry about him.

To guarentee myself the Peasant Jester semi-final the bracket would look like:

King vs. Jester

Peasant vs. Prince

Peasant vs. Jester, Peasant wins and I play him in the final.

In my bracket, I would play Earl first leaving me a 2/3 chance of winning, then I would let the Queen and Duke battle it out in the other round.

I will either face Queen (33% probability, and 50% chance of winning) or Duke (66% probability, 33% chance of winning ....

Wait light bulb....

If Queen and I are equally skilled, and Queen loses meaning duke succeeded and then I play him, wouldn't his probability of winning be adjusted to 50% since he already won 1 of the 2 out of 3 against my level? Or is this a really long tournament when players can take full rests and be 100% fresh for the next round?

If that is the chance, then I have a 67% in the quater final 50% in the semi, and 50% in the final leaving me with about 16.7% chance of winning

My group would be:

Queen vs. Duke

Me vs. Earl

Queen/Duke vs. Me

Me vs. Peasant

Edited February 18, 2008 by PolishNorbi

[unreality]

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

[unreality]

My own solution:

I know that I want this for A-D:

A: Jester

B: King

A wins

C: Peasant

D: Prince

C wins

but for the other 4, I have me (You), the Queen, the Duke and the Earl. How should I arrange it for maximum winning potential? Since there are four to arrange around, there are three scenarios:

You vs Duke --vs-- Queen vs Earl

You vs Queen --vs-- Duke vs Earl

You vs Earl --vs-- Duke vs Queen

*****

First Scenario

*****

Round One:

A: Jester

B: King

A wins

C: Peasant

D: Prince

C wins

E: You

F: Duke

1/3 chance you win

G: Queen

H: Earl

2/3 chance that Queen wins

Round Two:

Jester vs Peasant: Peasant wins

You/Duke/Duke vs Queen/Queen/Earl

To beat the Duke and the Queen your chances are 1/3 * 1/2 = 1/6

To beat the Duke and the Earl your chances are 1/3 * 2/3 = 2/9

You have a 2/3 chance of playing the queen and a 1/3 chance of playing the Earl, so your total chances of beating the Duke and then beating the winner of Queen vs Earl are:

(2/3 * 1/6) + (1/3 * 2/9) = 1/9 + 2/27 = 3/27 + 2/27 = 5/27

then you still have to beat the Peasant (half chance). So your final chances are 5/27 * 1/2 or 5/54

*****

Second Scenario

*****

Round One:

A: Jester

B: King

A wins

C: Peasant

D: Prince

C wins

E: You

F: Queen

1/2 chance you win

G: Duke

H: Earl

4/5 chance that Duke wins

Round Two:

Jester vs Peasant: Peasant wins

You/Queen vs Duke/Duke/Duke/Duke/Earl

Your chances of beating the Queen and the Duke are: 1/2 * 1/3 = 1/6

Your chances of beating the Queen and the Earl are: 1/2 * 2/3 = 1/3

You have a 4/5 chance of playing the Duke if you beat the Queen, and only a 1/5 chance of playing the Earl. So your chances of beating the Queen and then beating the winner of the Duke/Earl match are:

(4/5 * 1/6) + (1/5 * 1/3) = 2/15 + 1/15 = 3/15 = 1/5

But you still have to beat the Peasant, with a half chance. Thus your chances are 1/10 of winning the Tournament!

*****

Third Scenario

*****

Round One:

A: Jester

B: King

A wins

C: Peasant

D: Prince

C wins

E: You

F: Earl

2/3 chance you win

G: Duke

H: Queen

2/3 chance that the Duke wins

Round Two:

Jester vs Peasant: Peasant wins

You/You/Earl vs Duke/Duke/Queen

Your chances of beating the Earl and the Duke: 2/3 * 1/3 = 2/9

Your chances of beating the Earl and the Queen: 2/3 * 1/2 = 1/3

You have a 2/3 chance of playing the Duke and a 1/3 chance of playing the Queen. So your total chances of beating the Earl and then beating the winner of the Duke/Queen match are:

(2/3 * 2/9) + (1/3 * 1/3) = 4/27 + 1/9 = 4/27 + 3/27 = 7/27

You still have to beat the Peasant though. 7/27 * 1/2 = 7/54

****

The first scenario gave a 5/54 chance. 7/54 is obviously bigger than that.

7/54 = 12.96% chance of victory with the third scenario

1/10 = 10% chance of victory with the second scenario

Therefore, go with the third scenario, it's slightly better than the second

A: Jester

B: King

C: Peasant

D: Prince

E: You

F: Earl

G: Duke

H: Queen

let me know if you find any problems with this

As for multiple solutions:

The solution I gave was:

A: Jester

B: King

C: Peasant

D: Prince

E: You

F: Earl

G: Duke

H: Queen

There are actually many many solutions that give the same answer. You could flip the two halves, switch around the pairs inside their halves, and switch around the pairs themselves. So there are many same solutions, just with switched around brackets.

But by multiple solutions I meant better solutions. Ones with higher chances than 7/54. I'll keep looking

[unreality]

Any get a better chance than 7/54?

[Guest]

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.

[bonanova]

This puzzle is really a nice challenge. Kudos!!

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 ...

[1] compute individual playing strengths

Assign bonanova a strength of 100

Probability of bonanova beating the King = Bk = 0.25

Probability of the King beating bonanova = 1-Bk = 0.75

Strength ratio is ratio of winning probabilities:

S[King]/S[bonanova] = Sk/Sb = Kb/Bk = [1-Bk]/Bk = .75/.25 = 3

Strength[King] = Sk = 3 x Sb = 300

So:

Queen: 100 = 100 x [1-.50]/.50
Prince: 0 = 100 x [1-1 ]/1.0
Duke: 200 = 100 x [1-.33]/.33
Earl: 50 = 100 x [1-.66]/.66
Peasant: 100 = 100 x [1-.50]/.50
Jester: 0 - Can't compute Jester's strength from the info.
So take him out of the equation by pairing Jester against Peasant in round 1.
Against Peasant his strength is 0.

King:	300 = 100 x [1-.25]/.25

[2] Compute head-to-head winning probabilities Given relative strengths of the players, calculate the pairwise winning probabilities. We know that for bonanova and the King for example, Sk/Sb = [1-Bk]/Bk SkBk = Sb - SbBk Bk[sk+Sb] = Sb Bk = Sb/[sk + Sb] = 100/[300+100] = 0.25. OK Here are the rest of the head to head winning probabilities.

. | L O S E R |
+==========+=====+======+======+======+======+======+======+======+======+
| Winner | Str | bona | Jest | King | Duke | Queen| Peas | Earl | Prnce|
+==========+=====+======+======+======+======+======+======+======+======+
| bonanova 100 | x | 0 | .250 | .333 | .500 | .500 | .667 |1.000 |
| Jester - |1.000 | x | | | | 0 | | |
| King 300 | .750 | | x | .600 | .750 | .750 | .857 |1.000 |
| Duke 200 | .667 | | .400 | x | .667 | .667 | .800 |1.000 |
| Queen 100 | .500 | | .250 | .333 | x | .500 | .667 |1.000 |
| Peasant 100 | .500 |1.000 | .250 | .333 | .500 | x | .667 |1.000 |
| Earl 50 | .333 | | .143 | .200 | .333 | .333 | x |1.000 |
| Prince 0 | .000 | | .000 | .000 | .000 | .000 | .000 | x |
+==========+=====+======+======+======+======+======+======+======+======+

.				+======+======+======+======+======+======+======+======+

[3] Fill out brackets and get advancement probabilities

|
|---A---+
| +---A---+
|---B---+ |
| +---A---+
|---C---+ | |
| +---U---+ |
|---D---+ |
| |
| +--A--
| |
|---E---+ |
| +---V---+ |
|---F---+ | |
| +---X---+
|---G---+ |
| +---W---+
|---H---+

| Round 1 Round 2 Round 3

Compute probabilities for all entrants to reach the next three levels: Each entrant has a certain probability of winning Round 1 and reaching Round 2. Get those numbers directly from the table. Each entrant has a certain probability of winning Round 2 and reaching Round 3. Get those numbers from the Round 2 probabilities for all potential opponents and then from table and adding them up. Each entrant has a certain probability of winning Round 3 and being champion. Get those numbers from the Round 3 probabilities for all potential opponents and then from table and adding them up. e.g., Probabilities for A getting to Round 2: P[A2] = Ab Round 3: P[A3] = Ab.Au = P[A2].[P[C2].Ac + P[D2].Ad] Round 4: P[A4] = Ab.Au.Ax = P[A3].[P[E3].Ae + P[F3].Af + P[G3].Ag + P[H3].Ah]

[4] Best results. Strategy: Put bonanova in the top bracket [arbitrary] Put King/Duke in the bottom bracket where one of them will get knocked off Peasant/Jester knocks off Jester. Leaves only one choice - which bracket for Peasant/Jester match. [a] Peasant/Jester in top bracket:

100 1.000 .500 .172 - bonanova
0 .000 .000 .000 - Prince
100 1.000 .500 .172 - Peasant - same as bonanova
0 .000 .000 .000 - Jester
300 .600 .471 .354 - King - better than bonanova
200 .400 .284 .190 - Duke - better than bonanova
100 .667 .189 .094 - Queen
50 .333 .055 .018 - Earl

Strength Round 2 Round 3 Champion

Peasant/Jester in lower bracket: better because Earl might knock off Queen, getting me into semifinals with .556 prob instead of .500 above.

100 1.000 .556 .191 - bonanova 0.191 seems to be the best winning chance. Only the King's is better.
0 .000 .000 .000 - Prince
100 .667 .333 .114 - Queen
50 .333 .111 .024 - Earl
300 .600 .450 .343 - King <- better than .191
200 .400 .267 .182 - Duke
100 1.000 .283 .147 - Peasant
0 .000 .000 .000 - Jester

Strength Round 2 Round 3 Champion

[bonanova]

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. 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...!

[unreality]

The king is 3x better than you, correct? (since you can beat him 1/4 of the times). So can the Jester only beat the King 1/3 of the times? <-- that's what you're asking, right? Yeah, that can get tricky cuz you're not sure how good the Jester is in relation to everyone else.

I should have clarified about the Prince and Jester: Everyone beats the Prince. And the Jester beats everybody (excpet the Peasant of course)

and you're welcome for the riddle I have yet to look at your spoilers and see what you did to beat 7/54 (which I'm sure as possible, I just didnt have enough time lol)

edit: I just checked, nice strategy of assigning "strength points"

[bonanova]

The king is 3x better than you, correct? (since you can beat him 1/4 of the times). So can the Jester only beat the King 1/3 of the times? <-- that's what you're asking, right? Yeah, that can get tricky cuz you're not sure how good the Jester is in relation to everyone else.

I should have clarified about the Prince and Jester: Everyone beats the Prince. And the Jester beats everybody (excpet the Peasant of course)

and you're welcome for the riddle I have yet to look at your spoilers and see what you did to beat 7/54 (which I'm sure as possible, I just didnt have enough time lol)

edit: I just checked, nice strategy of assigning "strength points"

Just as a quick response, Jester will survive, and beat you, if you don't eliminate him using Peasant.

And when, other than Round 1 can you ensure the two will play each other?

You used Jester to eliminate King in R1, but that leaves Jester and Duke probably still alive.

I put Duke against King, and Peasant against Jester, to eliminate 2 of the 3 dangers [Jester, King, Duke] in R1,

and achieved a better than 50% survival rate into the finals.

Thanks... I'll go back and give Jester some insane strength number and redo some calculations.

This is intriguing!

[unreality]

What I did is I pitted the Jester against the King (guaranteed knockoff the hardest guy there, the King) and the Peasant against the Prince (ensuring that the Peasant moved on to play the Jester and beat him, before going up against me in the finals, giving me a 1/2 chance). Remember there are two absolutes- the Jester and the Prince (though the Prince has no exceptions, he just sucks at zarball )

[bonanova]

What I did is I pitted the Jester against the King (guaranteed knockoff the hardest guy there, the King) and the Peasant against the Prince (ensuring that the Peasant moved on to play the Jester and beat him, before going up against me in the finals, giving me a 1/2 chance). Remember there are two absolutes- the Jester and the Prince (though the Prince has no exceptions, he just sucks at zarball )

Yup .. I spoke before I thought. I'll run that simulation now.... thanks.

---------------------

Edit: I ran that simulation. It gives you a 0.130 winning chance.

The problem with that approach is that you have to fight through some strong opponents to reach the finals.

The good part is that you then have a 50% chance of winning the final match. IF you get there.

With You against Earl, you're in the semi's with a .667 probability.

But in the semi's you have a .667 probability of playing the Duke and a .333 probability of playing Queen.

That takes your odds of reaching the finals down to .259 so your winning chances are .130.

With my lineup:

----

Me - 100

Prince - 0

Queen - 100

Earl - 50

----

King - 300

Duke - 200

Peas - 100

Jest - 0 [loses to peas]

----

I get through Prince Queen and Earl - i.e. I make the finals with a .556 probability.

My opponent in the finals is:

King [against whom I have a .25 chance against his .450 probability of being there], or

Duke [against whom I have a .333 chance against his .267 probability of being there], or

Peasant [against whom I have a .5 chance against his .280 probability of being there], or

Jest [0 - he's dead]

My championship chances are thus:

.556 x [.25 x .450 + .333 x .267 + .5 x .280] = .556 x .341 = 0.191.

So I only have only a .341 chance of winning the final match,

but my .556 chance of making the finals more than offsets that.

[unreality]

I have a new strategy which I think might work pretty good, I'm testing it out now

[unreality]

My strategy:

Instead of fretting about everything else, I'm gonna look at what three people will give me maximum chance of victory:

The Prince, the Earl, and either the Queen or the Peasant, it doesnt matter

so I will pit the Prince against myself, the Earl against the Queen, and the Peasant against the Jester on the opposite side

Me vs Prince

Peasant vs Jester

Earl vs Queen

Duke vs King

with that I would have a half chance of making it to the finals (guaranteed victory against the Prince, 1/2 against the Peasant).

For the Earl vs Queen game, the Queen has a 2/3 chance of beating the Earl

For the Duke vs King game, the Duke is twice as good as me, the King is three times as good. So the King is 1.5x better than the Duke, meaning he has a 3/5 chance of beating the Duke, while the Duke has a 2/5 chance of beating the King.

So the combined chances of me beating the winner of the Earl/Queen game:

(2/3 * 1/2) + (1/3 * 2/3) = 1/3 + 2/9 = 5/9

the combined chances of me beating the winner of the Duke/King game:

(3/5 * 1/4) + (2/5 * 1/3) = 3/20 + 2/15 = 9/60 + 8/60 = 17/60

so those two (5/9 and 17/60) pitted against each other would be...

first match the chances' denominators: 100/180 and 51/180

those that are the chances that I will beat them. So I am 100:80 on the first one and 51:129 on the second one. I am 1.25x better than the first one, and 51/129 as good as the second one. Meaning the the first one is 204/645 as good as the second and has a 204/(204+645) chance of victory, or 204/849. So combined chances of me beating the winner of Earl vs Queen vs vs vs Duke vs King are:

(204/849 * 5/9) + (645/849 * 17/60) = .13349 + .21525 = .3486

or 34.86% chance of success Holy munchkins! That's really high!

wait nvm, I still have to fight the Peasant, so multiply that by 1/2:

17.43% chance

woohoo still pretty high! There could easily be an error in my ways though. Could anyone else check out this bracket:

Me vs Prince

Peasant vs Jester

Earl vs Queen

Duke vs King

[bonanova]

Confirm - you get to the finals with a .500 probability.

Your opponents are

Earl [.667 of beating him x.055 probability that he's in the finals] or

Queen [.5 x .189 prob] or

King [.25 x .471 prob] or

Duke [.333 x .284 prob]

Your chances of winning the final match is .3436 and being champ with half that - .171

We have differences in the 3rd decimal place, but I think we're getting the same result basically.

---------

It's still slightly better with Earl in your half of the bracket and make the finals with a .556 probability.

You than have a .341 chance of winning the final match and a .191 chance at the championship.

Which seems to make it advantageous to play weak opponents as long as possible

and then fight some statistical combination of the strong ones at the very end.

[unreality]

Oh yeah, I forgot that you had gotten that near-20%. woops lol. Yours is still the highest ;D it probably is the highest overall, though you never know

[unreality]

btw I'm working on another zarball riddle, that deals with the game itself

[Guest]

Can I at least get a little credit? I did post the same answer that Bonanova came up with long before the discussion between Unreality and Bonanova. Granted, I didn't do the math all the way out, but I was still right.

[unreality]

Wow you're right! Good job man! You got it first!

[Guest]

Thank you.

[bonanova]

You da man ... !