Group theory: Kingdom of Nashville

[Guest]

The King of Nashville decides to challenge his cabinet of 20 ministers. Each minister has to select a number in the range of 0 and 100 which is 50% of the most likely number selected by other minsters. The king announces that the group will be beheaded if anyone selects a wrong number or will go home with a booty of 100 coins each.

Each minister goes home rich, what number did they select and why?

[Guest]

This is an interesting one.

It is also a little hard to understand.

When you say 50% you mean half of the most likely chosen value?

Such that if the most likely chosen value is 50 then a minister should choose 25, right?

[Guest]

Yes, that is correct. As you pointed out if a minister thinks most people in the group will anticipate 50, then 50% of that number is 25.

The title of the puzzle should strike a bell, its got to do with some sort of equlibrium. Answer to be posted soon unless ofcourse someone figures it out!

[Guest]

The title wasn't speaking to me in a language that I understand. [please explain the title]

The answer should be 1.

[actually it should be 0 (edited)]

It's the only whole [cancel whole] number between 0-100 that doesn't have a half.

[1 has a half that can not be represented in whole #s, 0 has half and it can be represented by a # between 0-100 and that would be 0]

Since each minister has to pick half of the most likely # their pick will change "the most likely picked" number and there won't be a most likely picked # because it will keep changing.

[Guest]

Excellent, thats correct!

Title reflects the famous Nash equlibrium in Game Theory.

In a crude manner, in this problem, every minister would modify his action until and unless a further modification in his action will be no longer beneficial to himself and the group.

To start with number 25 (50% of 50) would be the most obvious choice. But knowing that everyone in the group knows this and hence they would select 25 means they would then select (50% of 25) and so on... until they reach 0. Hence 0 is the correct solution!

Will post a similar interesting puzzle in a few days!

[Guest]

To start with number 25 (50% of 50) would be the most obvious choice. But knowing that everyone in the group knows this and hence they would select 25 means they would then select (50% of 25) and so on... until they reach 0. Hence 0 is the correct solution!

either the answer is flawed or the question is.

the question says that if they pick the wrong number they will be beheaded or they will go home rich.

w=wrong number

b=beheaded

r=going home rich

w->(b v r)

therefore, if they picked any number but the right one they have an equal chance of going home rich or being beheaded. based on this formula, the fact that they went home rich does not lend to them having picked the correct answer and not the wrong one.

also, for a number to be a half of another number, it must double into that number it its half of. there is no number whole or otherwise that 0 is half of. 0 and 2 equals 0 still.

you did not specify whether the correct number is whole or otherwise, but regardless, a correct answer does not, by your question, automatically lend to going home rich.

theoretically, they could all have chosen different numbers and all gone home rich.

[Guest]

Nice puzzle, but wouldn't "Game Theory" be more appropriate than "Group Theory" in the title? Group theory refers to a specific branch of mathematics which doesn't seem to be the focus of this puzzle.

[Guest]

The King of Nashville decides to challenge his cabinet of 20 ministers. Each minister has to select a number in the range of 0 and 100 which is 50% of the most likely number selected by other minsters. The king announces that the group will be beheaded if anyone selects a wrong number or will go home with a booty of 100 coins each.

Each minister goes home rich, what number did they select and why?

m.txt

[Guest]

I totaliy disagree with the given solution.There is no logic.More close is to

say that the answer is 50,because The King said that everybody will take a 100

coins,if pik the right number.That may give the idea everybody to choose a 100

and 50% of a 100 is 50.But if you think logical the only sure number to pik is

0,because that is the only one number from which 50% gives the same number.As is

said everybody left with the award,that means thay are not so stuped and as i

said 0 is the only sure number.

[Guest]

0 is the answer. Because if everyone tries to get 50% of others number, no equilibrium will be there. The ultimate number for which each minister go home is 0. Otherwise someone at east should be beheaded.

[Guest]

The King of Nashville decides to challenge his cabinet of 20 ministers. Each minister has to select a number in the range of 0 and 100 which is 50% of the most likely number selected by other minsters. The king announces that the group will be beheaded if anyone selects a wrong number or will go home with a booty of 100 coins each.

Each minister goes home rich, what number did they select and why?

translate:

If wrong number then beheaded or 100 coins.

lemme make a truth table for you to show how this is flawed still

W=wrong number

R=going home rich

B=Beheaded

Premise:

If they choose a wrong number, then they will be beheaded or will go home with 100 coins.

Conclusion:

they went home with 100 coins

W ⊃ (B v R)

R.........B............W..............B v R.............W ⊃ (B v R)

t...........t.............t..................t.......................t

t...........t.............f..................t........................f

t...........f.............t..................t........................t

t...........f.............f..................t ........................ f

f...........t.............t..................t ........................t

f...........t.............f..................t ........................f

f...........f.............t..................f ........................f

f...........f.............f..................f ........................t

in rows 5 and 8 we have a true premise and a false conclusion. therefore, the logic of this question is not valid.

[Guest]

translate:

If wrong number then beheaded or 100 coins.

lemme make a truth table for you to show how this is flawed still

W=wrong number

R=going home rich

B=Beheaded

Premise:

If they choose a wrong number, then they will be beheaded or will go home with 100 coins.

Conclusion:

they went home with 100 coins

W ⊃ (B v R)

R.........B............W..............B v R.............W ⊃ (B v R)

t...........t.............t..................t.......................t

t...........t.............f..................t........................f

t...........f.............t..................t........................t

t...........f.............f..................t ........................ f

f...........t.............t..................t ........................t

f...........t.............f..................t ........................f

f...........f.............t..................f ........................f

f...........f.............f..................f ........................t

in rows 5 and 8 we have a true premise and a false conclusion. therefore, the logic of this question is not valid.

I think you did not get it, the precise statement is--"If one guesses wrong number, he will be beheaded, otherwise(if correct number) he will be rewarded 100 coins"...

hope you got it now.

[Guest]

I think you did not get it, the precise statement is--"If one guesses wrong number, he will be beheaded, otherwise(if correct number) he will be rewarded 100 coins"...

hope you got it now.

thats not what it says. my claim is that the question is flawed. as it is written, there is no right answer. beheaded if wrong number or go home rich does not translate into

wrong answer=beheaded

right= going home rich

as it is written, there is no stipulation for a right answer.

[Guest]

Well, not to pick nits, but the solution cannot be Zero because Zero is not BETWEEN Zero and One Hundred. If we limit the answer to whole numbers then the only options are 1 thru 99. Great Question Though.

[Guest]

The King of Nashville decides to challenge his cabinet of 20 ministers. Each minister has to select a number in the range of 0 and 100 which is 50% of the most likely number selected by other minsters. The king announces that the group will be beheaded if anyone selects a wrong number or will go home with a booty of 100 coins each.

Each minister goes home rich, what number did they select and why?

Undefined. "The king announces that the group will be beheaded if anyone . . . will go home with a booty of 100 coins." So, if they "win," they still get beheaded. It's a lose-lose situation.

[Guest]

The King of Nashville decides to challenge his cabinet of 20 ministers. Each minister has to select a number in the range of 0 and 100 which is 50% of the most likely number selected by other minsters. The king announces that the group will be beheaded if anyone selects a wrong number or will go home with a booty of 100 coins each.

Each minister goes home rich, what number did they select and why?

I think this depends on whether the king means 50% of the most likely figure ( which is theoretical), in which case 50 is likely to be the most popular. Or is the most likely figure the one that most minister actually choose. In this case each minister may choose 0 which would be 50% of the figure choosen by the other ministers

[Guest]

It doesnt seem entirely sound. The phrase 'likely number' is redundant. If they have to pick 50% of any number selected by another minister the result will be the same. The only way there can be a most likely number in this scenario is when they are all the same (ie 0). Essentially the King is saying "Ministers, choose the same number, and that number must also be equal to half of itself". It's kind of dumb

[Guest]

thats not what it says. my claim is that the question is flawed. as it is written, there is no right answer. beheaded if wrong number or go home rich does not translate into

wrong answer=beheaded

right= going home rich

as it is written, there is no stipulation for a right answer.

Why don't you just try to answer the question as it is intended instead of trying to nullify it. Clearly, the author wanted you to answer a riddle, not confirm that he wrote a flawed question.

I think the King had something going there, because he wanted to make sure he didn't have a bunch of idiot ministers. He wanted them to come up with a number, X, so that X=X/2. 0 is the only possible solution obviously. If they all answer correctly, they get money, if even one answers incorrectly, they all die.

[Guest]

@ Wiiosten

Seriously why on every question that is slightly flawed do people have to point it out and do alot of extra work to prove that the question is flawed?

If you have enough sense to prove that it's flawed you should fairly easily be able to deduce what he meant to ask, why not mike a nice post saying oh i think you meant to say ... whatever.. and then answer that instead of wasting your time doing a bunch of unnecessary work to make yourself sound like an arrogant know-it-all who thinks he's always right?

/gets off his soap box

and for the answer yeah I probably wouldn't have gotten it but it makes since now that i read all the posts =)

[Guest]

wouldnt it b 0 cause if every1 was considered smart then they would think as follows:

1/2 of 100 is 50 but since they will realize this...

1/2 of 50 is 25

and so on

anyway they would eventually get to 0.

[Guest]

0 is the answer. Because if everyone tries to get 50% of others number, no equilibrium will be there. The ultimate number for which each minister go home is 0. Otherwise someone at east should be beheaded.

technically it is impossible to select 0 as the answer since the question states "between 0 and 100"

Edited April 14, 2009 by eezook

[Guest]

The question is definitely flawed but not for the reason mentioned previously.

The question is based on the ASSUMPTION that there exist a "most likely chosen number".

But that assumption may not be correct.

If all the 20 ministers select different numbers than there is no "most likely chosen number" and there is no question of choosing half of most likely chosen number.

And in that case King is in trouble because he has assumed that there will be a most likely chosen number.

Question makes more sense if ministers are at least 100 or more than 100.

Because in that case it is sure that there exists a "most likely chosen number".

[benjer3]

I agree with pktrivedi in that if all of the ministers chose a different number, the king's riddle would not work. However, as BigTime pointed out, the king was probably testing his ministers. The obvious solution was zero, and if anyone didn't pick that, the cabinet would be replaced.

For those who focused on the wording, the condition "if anyone selects a wrong number" only defines the consequence "the group will be beheaded," and the implied "otherwise" defines the consequence "[the group] will go home with a booty of 100 coins each." The "or" separates these. The sentence isn't questionable.

And like K4D said, the point of this forum is to solve the given riddles, not to point out minor flaws and loopholes. As long as the meaning is understood and a reasonable, valid answer can be deduced, the point should be to find the answer. I think it's okay to point out flaws as long as the answer is acknowledged and as long as it is done good-heartedly rather than derisively.

Edited January 6, 2011 by benjer3

[Guest]

what do you mean by "Each minister has to select a number in the range of 0 and 100 which is 50% of the most likely number selected by other minsters"?

[Quantum.Mechanic]

Well, not to pick nits, but the solution cannot be Zero because Zero is not BETWEEN Zero and One Hundred. If we limit the answer to whole numbers then the only options are 1 thru 99. Great Question Though.

Questions using "between", giving an interval, are generally intepreted as using the closed interval, thus 0-100 means [0-100], including the endpoints. Sure, the language is imprecise, and much confusion has occurred in the past when the open interval is intended. This is a bigger problem for real (non-integer) ranges, at least in many people's minds -- it seems natural to include the endpoints when stating an integer range, less so when giving a non-integer range.

Also reminds me of the language issue with things like "under-10" kids leagues, where the endpoint is not included (and it is a non-integer range).

[fabpig]

mmmmmm. Hate to be a spoilsport, fellas, but most of this thread was done 2 or 3 years ago and the protagonists long gone. So you're not going to get any replies.....