I know you probably got a lot of feedback on this one, but i think something is missing from the answer/result for people to understand.
What i get from the question is
* You do not know what the red/blue ratio of hats are.
* You do not know if the person(s) behind you survived or not - since they are killed *silently* and if they survive must "remain absolutely silent".
The answer
If 99 are guaranteed to survive only one person can die. The first wise to be asked the question have nothing to base his answer on - regardless of the number of red and blue in front of him, his hat can be any color. His answer will be based entirely on the formular you mentioned thus saving the man in front of him. And a 50% chance of saving himself as well.
I think the real kicker here to understand is the fact that the one person not guaranteed to survive is the first. Your two examples have the first man to call out his own hats color by sheer luck - a better example #2 would be of him making a mistake and dying.
Also it's worth mentioning, that if the first man (alone) somehow fails to call the right color - everybody dies.
The following puzzle posted on ur website does not give correct answer.
Consider the scenario:
order of being questioned by the king is from 9 to 1.
->
R B B R B B B R R
9 8 7 6 5 4 3 2 1
9 sees odd red hats, says Blue and is killed. (The first one has 50% chance to live anyways)
8 sees odd red hats, says Blue
7 sees odd red hats, says Blue
6 sees even red hats, says Red
5 sees even red hats, says Red and is killed !!!!!!
This shows that 99 ppl cannot be saved and the logic is erronous.
The trick is to switch the code for odd and even red hats everytime there is change of color in an answer.
Consider the scenario:
order of being questioned by the king is from 9 to 0.
->
R B B R B B B R R B
9 8 7 6 5 4 3 2 1 0
9 sees odd red hats, says Blue and is killed. (The first one has 50% chance to live anyways)
8 sees odd red hats, says Blue
7 sees odd red hats, says Blue
6 sees even red hats, says Red (change of color in answer)
5 sees even red hats, says Blue!! (hence switch code)
4 sees even red hats, says Blue
3 sees even red hats, says Blue
2 sees odd red hats, says Red (again change of color in answer)
1 sees even red hats, says Red!! (hence switch code)
that way all 99 will survive.. now we know why there were "WISE" men!!
The guy who is questioned first will tell the colour opposite to the color of the hat of the guy in front him.(If the guy in front of him has a red hat he would say blue).This would be the code and the guy in front of him would understand that the color of his own hat is red and thus from there on all would be saved. The reason why the first guy would announce the opposite color is because the king would have sensed the normal plan of the first guy announcing the color of the hat in front of him as an expected thing and wouldn't have the same color for the last two guys.
The answer to this puzzle, (as with most), is very simple. First, we don't know the ratio of red hats to blue hats, so I don't think a mathmatical answer is possible. Here's my solution:
The wise man in the back simply says the color of the hat in front of him, and he has a 50% of living. Now the next wise man knows the color of his own hat. If his hat is the same color of the wise man in front of him, he calls out the color just loud enough to ensure the wise man in front of him hears it. If his hat is a different color than the wise man in front of him, he yells out the color and so on down the line.
The king never restricted intonation, and they are simply calling out the color of their own hat.
I think you didn't get the answer. Doesn't matter..here is how it works for the example you gave:
The Q He Reds Successful
says In Front Reds
R B X X - KILLED in this case
R R Even Even
R R Odd Odd
R R Even Even
B B Even Odd
B B Even Odd
R R Odd Odd
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
R R Even Even
B B Even Odd
R R Odd Odd
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
R R Even Even
R R Odd Odd
R R Even Even
If you still don't get it.. Well..go through the solution details again.
there is no need to make this harder than it is. 1 wise man has to die (or runs a 50% chance of dying anyway) and its the last in the queue. basically, he calls 'red' if he sees that the number of red hats is odd, and blue if he sees that the number of blue hats is odd. as there are 99 wise men in front of him, it will always be the case that one colour will have an odd number and the other an even number. each person simply counts the number of red and blue hats in front of them as well as recalls the number of red and blue hats announced behind them. it will either be odd/odd or even/even. based on the clue given by the last person, each can decide if the hat he is wearing is red or blue. eg: if the clue reveals that the number of red hats is odd, and a particular wise man down the line registers an even number of red hats (both in front and behind him), then his hat has to be red.
In most logical problems, if a ratio isn't mentioned, the Rule of Thumb is to assume it is fifty fifty. Now, the first man has a 50% chance of dying. If he sees that the blues are odd, him being a wise man, he will say blue and vice versa. The man in front of him follows the same rule until all of them are done. If one person dies, it will be the very first one. Simple ^-^
maybe they should use some kind of system like knowing what how the other wise men think when picking 1 of 2 colors. if one of the wise men picks a color based on his personality or what color is closest to his favorite color maybe that should work
We'll call the first wise man 100, the second 99, the third 98, and so on.
100 sees 99 hats in front of him. Since this is odd, there must be an odd number of one color and an even number of the other. He will call out the color that is odd. It's possible that he calls the wrong color so he can't be counted as a guaranteed survivor.
Assume 100 calls out blue.
Now all the wise men know that blue is odd, red is even.
Suppose 99 sees that red is odd, he knows its value changed so his hat is red. He calls out red, now everyone knows the value of red changed to odd and blue is still odd.
Now suppose 98 sees that blue is even, he knows its value changed so his hat is blue. He calls out blue, now everyone knows the value of blue changed to even and red is still odd.
Now suppose 97 sees that blue is odd, he knows that its value changed so his hat is blue. He calls our blue, now everyone knows the value of blue changed to odd and red is still odd.
The ratio of blue to red isn't important. The first wise man calls out the odd color and after that it is a matter of listening to which color changes from odd to even.
wise man calls The wise man sees
100 blue blue-odd red-even
99 blue-even red-even
98 blue-odd red-even
97 blue-odd red-odd
96 blue-odd red-even
95 blue-odd red-odd
94 blue-even red-odd
Each wise man sees a certain number of blue and red hats. The first wise man sees one odd color and one even color. For the rest of the wise men, the number of red and blue hats could both be odd, both even, or one could be odd and one could be even. The wise man are calling out what has changed from odd to even or even to odd for their turn. Each time a wise man calls out a color, the rest of the wise men know that call has just changed from odd to even or even to odd.
wise man calls The wise man sees
100 blue blue-odd red-even
99 blue blue-even red-even
98 blue blue-odd red-even
97 red blue-odd red-odd
96 red blue-odd red-even
95 red blue-odd red-odd
94 blue blue-even red-odd
In this case it started out with blue odd, red even
Then 99 saw that blue had changed from odd to even and called out blue. Now everyone knows that blue is even and red is even.
Next 98 saw that blue had changed from even to odd and called out blue. Now everyone knows that blue is odd and red is even.
Each wise man has to keep track of the odd-even state of the number of hats for every turn. This is because only the first color called was associated with being odd. All other colors called are stating that the color has changed from odd to even or even to odd, but you can't know which way it switched unless you know every single change that is made from the starting point.
Also, it needs to be noted that for the example "Doesnt Matter" gave, this doesn't work. There needs to be an even number of wise men. This ensures that the first wise man has an odd number of people in front of him. The color he calls signifies that color is odd and the other is even. If he has an even number of people in front of him there won't be an odd number of one color and an even number of the other. There would either be an even number of red and an even number of blue or an there would be an odd number of red and an odd number of blue.
There's an easy way for the wise men to keep track of the odd-even state of the colors. If the first wise man calls out blue, then everyone makes the mental note,
R=0 B=1.
Now suppose the following colors are called out:99R, 98R, 97B, 96R, 95R, 94R, 93B, 92B, 91R, 90B, 89B, 88R, 87R, 86R, 85B, 84B
Wise man 83 counts the number of reds and blues, then adds them to the total of R=0 and B=1.
There are 9R and 7B, so R=0+9 and B=1+7, R=9 and B=8.
Since R is odd and B is even, wise man 83 knows that including his hat, there are an odd number of red hats and an even number of blue hats. If he sees even red hats, then his hat is red. If he see odd red hats, then his hat is blue.
In the discussion they should decide that the first persion tobe asked should call the colour he found as a odd no. and this way others by keeping the track of the colour will be able to call their colours successfully.
Only 50% of the people are guaranteed to stay alive.
100th men will call the colour of the Hat in front of him.
So 99th men will just call the colour said by the 100th men.
In this way all men 99,97,95 etc are sure to live.
but men in the position 100,98,96 etc are not sure of living
So 50%of the men are guaranteed that they will live.
Guys, this is far simpler than you are making it. The question asks "what is the MAXIMUM number of people who are guarenteed to be saved?" - well there is only 1 wise man that cannot be guarenteed safety and that is the first wise man.
It is not asking how many will be guarenteed - rather what is the maximum number that can be guarenteed. Regardless of the system they adopt to guess correctly - only person number 1 has no possible way of guarenteeing his safety
The safety of none of the wise men is gauranteed. Consider this:
the 100th wise man must see a pattern of even and odd sets (ex: 98 red, 1 blue-even,odd). Naturally, he has no way of determining the color of his own hat. The 99th man must see a pattern of either even and even or odd and odd sets. Now his hat could be either red or blue, as either case would satisfy the even and odd orientation which the man before him sees, and he has no basis for judgement as he does not know the color of the man behind him.
look one is infront of the other so number 100 is going to get killed because he does not know his color but yet he says all of the others colored hats and each learns the number red and blue colored hats since #100 wise guy was killed he doesnt count anymore just those who remain meaning only 1 died because he answered his color wrong but looked at the rest of the ppl and counted them all remebered the number
but seriouslly, the king got bored and need something to do.
he can always have sex with all the female concubian he have rather that playing such a deadly game.
What i get from the question is
* You do not know what the red/blue ratio of hats are.
* You do not know if the person(s) behind you survived or not - since they are killed *silently* and if they survive must "remain absolutely silent".
The answer
If 99 are guaranteed to survive only one person can die. The first wise to be asked the question have nothing to base his answer on - regardless of the number of red and blue in front of him, his hat can be any color. His answer will be based entirely on the formular you mentioned thus saving the man in front of him. And a 50% chance of saving himself as well.
I think the real kicker here to understand is the fact that the one person not guaranteed to survive is the first. Your two examples have the first man to call out his own hats color by sheer luck - a better example #2 would be of him making a mistake and dying.
Also it's worth mentioning, that if the first man (alone) somehow fails to call the right color - everybody dies.
Great puzzle selection btw!
order of being questioned by the king is from 9 to 1.
->
R B B R B B B R R
9 8 7 6 5 4 3 2 1
9 sees odd red hats, says Blue and is killed. (The first one has 50% chance to live anyways)
8 sees odd red hats, says Blue
7 sees odd red hats, says Blue
6 sees even red hats, says Red
5 sees even red hats, says Red and is killed !!!!!!
This shows that 99 ppl cannot be saved and the logic is erronous.
Consider the scenario:
order of being questioned by the king is from 9 to 0.
->
R B B R B B B R R B
9 8 7 6 5 4 3 2 1 0
9 sees odd red hats, says Blue and is killed. (The first one has 50% chance to live anyways)
8 sees odd red hats, says Blue
7 sees odd red hats, says Blue
6 sees even red hats, says Red (change of color in answer)
5 sees even red hats, says Blue!! (hence switch code)
4 sees even red hats, says Blue
3 sees even red hats, says Blue
2 sees odd red hats, says Red (again change of color in answer)
1 sees even red hats, says Red!! (hence switch code)
that way all 99 will survive.. now we know why there were "WISE" men!!
The wise man in the back simply says the color of the hat in front of him, and he has a 50% of living. Now the next wise man knows the color of his own hat. If his hat is the same color of the wise man in front of him, he calls out the color just loud enough to ensure the wise man in front of him hears it. If his hat is a different color than the wise man in front of him, he yells out the color and so on down the line.
The king never restricted intonation, and they are simply calling out the color of their own hat.
No matter what the formula, it wont work unless you say your answer unnormal.
What i mean by this is if u say it quiet,loud, fast etc. *cheating*
The king will hear the people before they line up discussing how they will say their answer.
Unless your idea works on these first 25 people dont question me.
R R R R B B R B B B B B B R B R B B B B B B R R R
I think you didn't get the answer. Doesn't matter..here is how it works for the example you gave:
The Q He Reds Successful
says In Front Reds
R B X X - KILLED in this case
R R Even Even
R R Odd Odd
R R Even Even
B B Even Odd
B B Even Odd
R R Odd Odd
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
R R Even Even
B B Even Odd
R R Odd Odd
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
B B Odd Even
R R Even Even
R R Odd Odd
R R Even Even
If you still don't get it.. Well..go through the solution details again.
All the best.
Cheers!
100 sees 99 hats in front of him. Since this is odd, there must be an odd number of one color and an even number of the other. He will call out the color that is odd. It's possible that he calls the wrong color so he can't be counted as a guaranteed survivor.
Assume 100 calls out blue.
Now all the wise men know that blue is odd, red is even.
Suppose 99 sees that red is odd, he knows its value changed so his hat is red. He calls out red, now everyone knows the value of red changed to odd and blue is still odd.
Now suppose 98 sees that blue is even, he knows its value changed so his hat is blue. He calls out blue, now everyone knows the value of blue changed to even and red is still odd.
Now suppose 97 sees that blue is odd, he knows that its value changed so his hat is blue. He calls our blue, now everyone knows the value of blue changed to odd and red is still odd.
The ratio of blue to red isn't important. The first wise man calls out the odd color and after that it is a matter of listening to which color changes from odd to even.
wise man calls The wise man sees
100 blue blue-odd red-even
99 blue-even red-even
98 blue-odd red-even
97 blue-odd red-odd
96 blue-odd red-even
95 blue-odd red-odd
94 blue-even red-odd
Each wise man sees a certain number of blue and red hats. The first wise man sees one odd color and one even color. For the rest of the wise men, the number of red and blue hats could both be odd, both even, or one could be odd and one could be even. The wise man are calling out what has changed from odd to even or even to odd for their turn. Each time a wise man calls out a color, the rest of the wise men know that call has just changed from odd to even or even to odd.
wise man calls The wise man sees
100 blue blue-odd red-even
99 blue blue-even red-even
98 blue blue-odd red-even
97 red blue-odd red-odd
96 red blue-odd red-even
95 red blue-odd red-odd
94 blue blue-even red-odd
In this case it started out with blue odd, red even
Then 99 saw that blue had changed from odd to even and called out blue. Now everyone knows that blue is even and red is even.
Next 98 saw that blue had changed from even to odd and called out blue. Now everyone knows that blue is odd and red is even.
Each wise man has to keep track of the odd-even state of the number of hats for every turn. This is because only the first color called was associated with being odd. All other colors called are stating that the color has changed from odd to even or even to odd, but you can't know which way it switched unless you know every single change that is made from the starting point.
R=0 B=1.
Now suppose the following colors are called out:99R, 98R, 97B, 96R, 95R, 94R, 93B, 92B, 91R, 90B, 89B, 88R, 87R, 86R, 85B, 84B
Wise man 83 counts the number of reds and blues, then adds them to the total of R=0 and B=1.
There are 9R and 7B, so R=0+9 and B=1+7, R=9 and B=8.
Since R is odd and B is even, wise man 83 knows that including his hat, there are an odd number of red hats and an even number of blue hats. If he sees even red hats, then his hat is red. If he see odd red hats, then his hat is blue.
100th men will call the colour of the Hat in front of him.
So 99th men will just call the colour said by the 100th men.
In this way all men 99,97,95 etc are sure to live.
but men in the position 100,98,96 etc are not sure of living
So 50%of the men are guaranteed that they will live.
It is not asking how many will be guarenteed - rather what is the maximum number that can be guarenteed. Regardless of the system they adopt to guess correctly - only person number 1 has no possible way of guarenteeing his safety
the 100th wise man must see a pattern of even and odd sets (ex: 98 red, 1 blue-even,odd). Naturally, he has no way of determining the color of his own hat. The 99th man must see a pattern of either even and even or odd and odd sets. Now his hat could be either red or blue, as either case would satisfy the even and odd orientation which the man before him sees, and he has no basis for judgement as he does not know the color of the man behind him.
but seriouslly, the king got bored and need something to do.
he can always have sex with all the female concubian he have rather that playing such a deadly game.