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* * * * * 1 votes

For Bushindo - (others try at your own risk)


Best Answer Prime, 15 March 2013 - 10:19 PM

I am uncertain, what uncertain means in this context.

Cole's first shot strategy is clear. He must shoot himself in the foot thus exiting the duel with a non-fatal injury. (Hopefully, he does not miss.)

Spoiler for the honorable way

Go to the full post


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18 replies to this topic

#11 Prime

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Posted 18 March 2013 - 10:32 AM

Let me try and bring consensus here by wearing everyone down with a lengthy, unnecessarily tedious, and tiresome detailed solution.

Outside of solving inequality, which I think is more revealing than equality, I don't see any significant difference in the results. I did use a little shortcut without giving an adequate explanation/justification for it. Therefore, I feel compelled to clarify the point where the reasoning ends and algebra begins.

Spoiler for detailed solution

Hopefully, this solves all tiebreak situations and removes the uncertainty.

 


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Past prime, actually.


#12 bonanova

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Posted 18 March 2013 - 04:12 PM

Let me try and bring consensus here by wearing everyone down with a lengthy, unnecessarily tedious, and tiresome detailed solution.

Outside of solving inequality, which I think is more revealing than equality, I don't see any significant difference in the results. I did use a little shortcut without giving an adequate explanation/justification for it. Therefore, I feel compelled to clarify the point where the reasoning ends and algebra begins.

Spoiler for detailed solution

Hopefully, this solves all tiebreak situations and removes the uncertainty.

 

Reading from the OP:

Given that it [Cole's strategy] is uncertain, what can we determine regarding Bobby's shooting accuracy?

 

From your reasoning regarding Cole's strategy, what is the answer?


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

#13 Prime

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Posted 18 March 2013 - 04:39 PM

Let me try and bring consensus here by wearing everyone down with a lengthy, unnecessarily tedious, and tiresome detailed solution.

Outside of solving inequality, which I think is more revealing than equality, I don't see any significant difference in the results. I did use a little shortcut without giving an adequate explanation/justification for it. Therefore, I feel compelled to clarify the point where the reasoning ends and algebra begins.

Spoiler for detailed solution

Hopefully, this solves all tiebreak situations and removes the uncertainty.

 

Reading from the OP:

Given that it [Cole's strategy] is uncertain, what can we determine regarding Bobby's shooting accuracy?

 

From your reasoning regarding Cole's strategy, what is the answer?

We are arguing semantics. To me uncertain means cannot be determined, like division by zero. I just do not see the choice between two equally good (bad) values as an uncertainty. And I have given several good ways to decide the tiebreak. I don't think it's all that important in our three way duel. I figure, Bushindo's objection was to omitting inclusion of all of the variations into the equation. I think we are in the right to do that. And after algebraic simplifications the full equations would come to the same thing. And we all use different nomenclature. (Mine is the easiest to type, although not as formal.)

I found it interesting that in Bonanova's analysis, using probability of missing (q), simplifies the solution. In particular the solution of the cubic equation.

I say, this duel is solved.

Alternatively, shooting himself in the foot may be the best option for Cole.


Edited by Prime, 18 March 2013 - 04:41 PM.

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Past prime, actually.


#14 bushindo

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Posted 18 March 2013 - 04:59 PM

Let me try and bring consensus here by wearing everyone down with a lengthy, unnecessarily tedious, and tiresome detailed solution.

Outside of solving inequality, which I think is more revealing than equality, I don't see any significant difference in the results. I did use a little shortcut without giving an adequate explanation/justification for it. Therefore, I feel compelled to clarify the point where the reasoning ends and algebra begins.

Spoiler for detailed solution

Hopefully, this solves all tiebreak situations and removes the uncertainty.

 

Reading from the OP:

Given that it [Cole's strategy] is uncertain, what can we determine regarding Bobby's shooting accuracy?

 

From your reasoning regarding Cole's strategy, what is the answer?

 

 

I agree about semantics. I think the underlying crux of the discussion is the interpretation of 'uncertain'

Spoiler for


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#15 Prime

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Posted 18 March 2013 - 05:33 PM

 

Let me try and bring consensus here by wearing everyone down with a lengthy, unnecessarily tedious, and tiresome detailed solution.

Outside of solving inequality, which I think is more revealing than equality, I don't see any significant difference in the results. I did use a little shortcut without giving an adequate explanation/justification for it. Therefore, I feel compelled to clarify the point where the reasoning ends and algebra begins.

Spoiler for detailed solution

Hopefully, this solves all tiebreak situations and removes the uncertainty.

 

Reading from the OP:

Given that it [Cole's strategy] is uncertain, what can we determine regarding Bobby's shooting accuracy?

 

From your reasoning regarding Cole's strategy, what is the answer?

 

 

I agree about semantics. I think the underlying crux of the discussion is the interpretation of 'uncertain'

Spoiler for

Now Bushindo is with Bonanova insisting that Win(shooting air)=Win(shooting Alex) means uncertainty. What's wrong with my tiebreaks?

A true 3-way duel indeed. But all got the same answer to the problem.

I can see where my guidlines to Cole in post#4 could be misinterpreted. My intention was to help Cole avoiding computations, where possible. For if he sits down with a calculator in the middle of the duel, the other two guys may get angry and just shoot him out of turn.


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Past prime, actually.


#16 bushindo

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Posted 18 March 2013 - 07:31 PM

Now Bushindo is with Bonanova insisting that Win(shooting air)=Win(shooting Alex) means uncertainty. What's wrong with my tiebreaks?

A true 3-way duel indeed. But all got the same answer to the problem.

I can see where my guidlines to Cole in post#4 could be misinterpreted. My intention was to help Cole avoiding computations, where possible. For if he sits down with a calculator in the middle of the duel, the other two guys may get angry and just shoot him out of turn.

 

 

I think I misinterpreted your position. I don't think bononova and I share the same interpretation of 'uncertain', though. Let me see if we have the following positions correct

Spoiler for


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#17 Prime

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Posted 18 March 2013 - 09:47 PM

Now Bushindo is with Bonanova insisting that Win(shooting air)=Win(shooting Alex) means uncertainty. What's wrong with my tiebreaks?

A true 3-way duel indeed. But all got the same answer to the problem.

I can see where my guidlines to Cole in post#4 could be misinterpreted. My intention was to help Cole avoiding computations, where possible. For if he sits down with a calculator in the middle of the duel, the other two guys may get angry and just shoot him out of turn.

 

 

I think I misinterpreted your position. I don't think bononova and I share the same interpretation of 'uncertain', though. Let me see if we have the following positions correct

Spoiler for

The way I see it, Bonanova has made his problem statement/position 100% clear:

Uncertainty is when Cole's winning chance by shooting in the air = his winning chance by shooting at Alex.

The question Bonanova wants us to solve is: For what values of b (accuracy of Bobby) such situation is possible?

And solved it we have:

Spoiler for the answer

Bushindo's view of uncertainty is the same as Bonanova's: Wshoot_air(b,c) = Wshoot_at_Alex(b,c).

However, there seems to be some uncertainty as to what exactly Bonanova wants us to find.

Prime picks on the usage of the word uncertain. Does not believe there is any uncertainty here at all. (When chances are equal, Cole must shoot Alex, because he hates him more than he hates Bobby.) Also, Prime promotes (unsuccessfully) an alternative strategy whereby Cole shoots himself in the foot with his very first shot.


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Past prime, actually.


#18 bushindo

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Posted 19 March 2013 - 02:19 AM

 

Now Bushindo is with Bonanova insisting that Win(shooting air)=Win(shooting Alex) means uncertainty. What's wrong with my tiebreaks?

A true 3-way duel indeed. But all got the same answer to the problem.

I can see where my guidlines to Cole in post#4 could be misinterpreted. My intention was to help Cole avoiding computations, where possible. For if he sits down with a calculator in the middle of the duel, the other two guys may get angry and just shoot him out of turn.

 

 

I think I misinterpreted your position. I don't think bononova and I share the same interpretation of 'uncertain', though. Let me see if we have the following positions correct

Spoiler for

The way I see it, Bonanova has made his problem statement/position 100% clear:

Uncertainty is when Cole's winning chance by shooting in the air = his winning chance by shooting at Alex.

The question Bonanova wants us to solve is: For what values of b (accuracy of Bobby) such situation is possible?

And solved it we have:

Spoiler for the answer

Bushindo's view of uncertainty is the same as Bonanova's: Wshoot_air(b,c) = Wshoot_at_Alex(b,c).

However, there seems to be some uncertainty as to what exactly Bonanova wants us to find.

Prime picks on the usage of the word uncertain. Does not believe there is any uncertainty here at all. (When chances are equal, Cole must shoot Alex, because he hates him more than he hates Bobby.) Also, Prime promotes (unsuccessfully) an alternative strategy whereby Cole shoots himself in the foot with his very first shot.

 

 

You're right. My view of uncertainty is the same as bonanoba's. It's still too early for me to be senile, *sigh*.

 

Good analysis, though. Your analysis definitely deserves the distinction of Best Answer.


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#19 bonanova

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Posted 19 March 2013 - 04:41 AM

Most people attribute the "truel" puzzle to Martin Gardner.

 

Its charm, like many of his puzzles, it that doing the thinking

and the math rewards you with a surprise. The surprise here

is that the best shooter can end up the underdog, and the
worst shooter
can end up the favorite to win.

 

If Alex shoots first we'll bet on him all day long. Having to shoot

last is clearly the reason why he might be least likely to win.

And that is not a greatly surprising result.

 

But it is not because Cole may shoot first that he can be the

most likely to win: it's because he is able to "volunteer" to

shoot last -- along with his being the least desirable target.

That is the genius of Gardner.

I can't add anything to what you guys have said.

This was enjoyable.


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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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