absoluzation
New member
- Joined
- Dec 4, 2019
- Messages
- 27
QUESTION:
As the tradition demands, Santa has once again invited some of his smartest elves for coffee and cake this year. The ten clever elves are already excited when Santa finally shows up.
“Please excuse me for being late,” he calls out, panting, “I just had to get a few more hats. All right then, let's get started: My dear clever elves! As every year, I would like to give you the chance to win coffee and cake. All you have to do is guessing the colour of the hat on your head.”
“Red!” Egobert shouts.
Santa laughs: “Of course, only after I have put the hats on your head. The game is very simple: first, you line up in a row. Please make sure that each of you can only see the people in front of you. This means that if Albert is standing at the back, he will see all the others: Bertha, Claudio and so on up to Julia. But Bertha, who is standing second to last, only sees Claudio up to Julia, but not Albert, understand? And don't you dare cheating! As soon as you've all lined up, you'll each be given either a red or a green hat—such that you can't see them yourself, of course. Then you have to guess the colour of your hat one after another. Albert (who is at the back) starts and says aloud either ‘Red!’ or ‘Green!’. Then it's Bertha's turn, then Claudio's and so on. At the end, I will tell you how many of you have guessed their hat colour correctly. The more you guess correctly, the more pieces of cake you will get.”
“Are we allowed to discuss?” Franka asks.
“For now, you are welcome to discuss a strategy. However, as soon as I start putting the hats on your heads, I don't want to hear the slightest peep. You are only allowed to say ‘Red!’ or ‘Green!’ once when it's your turn. And please don't try to communicate any more information. Whether it's by how you communicate your guess or how long you wait or something like that. Because then I would have to eat the whole cake all by myself...”
He grievingly glances down at his round Santa belly and asks: “Do you have any further questions?”
“Are the hats chosen at random?” Immanuel asks.
“Yes indeed, I will choose the colour of each hat completely at random.”
After some thought, the elves come up with a strategy that will guarantee them as much cake as possible. What is the maximum number of elves who are guaranteed to guess their hat colour correctly?
Possible answers:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
MY ANSWER:
For each person who gets told their hat color, someone has to waste their opportunity to guess the color of their own hat for someone else.
For example: There are 10 elves in total. The first elf doesn't know their hat color, so he shouts the tenth elf's hat color. The same goes for elves 2 and 9, 3 and 8, 4 and 7, 5 and 6. This means elves six to ten are guaranteed to guess their hat color correctly, whereas it's not guaranteed whether elves one to five will be able to guess theirs correctly. This means the maximum number of the elves who are guaranteed to guess their hat color is 5.
So I've asked this in a different forum as well and the answer is probably over 5. Could someone tell me how to solve this and give me an explanation as to why it is the right answer? It would mean a lot.
Thank you for your time.
As the tradition demands, Santa has once again invited some of his smartest elves for coffee and cake this year. The ten clever elves are already excited when Santa finally shows up.
“Please excuse me for being late,” he calls out, panting, “I just had to get a few more hats. All right then, let's get started: My dear clever elves! As every year, I would like to give you the chance to win coffee and cake. All you have to do is guessing the colour of the hat on your head.”
“Red!” Egobert shouts.
Santa laughs: “Of course, only after I have put the hats on your head. The game is very simple: first, you line up in a row. Please make sure that each of you can only see the people in front of you. This means that if Albert is standing at the back, he will see all the others: Bertha, Claudio and so on up to Julia. But Bertha, who is standing second to last, only sees Claudio up to Julia, but not Albert, understand? And don't you dare cheating! As soon as you've all lined up, you'll each be given either a red or a green hat—such that you can't see them yourself, of course. Then you have to guess the colour of your hat one after another. Albert (who is at the back) starts and says aloud either ‘Red!’ or ‘Green!’. Then it's Bertha's turn, then Claudio's and so on. At the end, I will tell you how many of you have guessed their hat colour correctly. The more you guess correctly, the more pieces of cake you will get.”
“Are we allowed to discuss?” Franka asks.
“For now, you are welcome to discuss a strategy. However, as soon as I start putting the hats on your heads, I don't want to hear the slightest peep. You are only allowed to say ‘Red!’ or ‘Green!’ once when it's your turn. And please don't try to communicate any more information. Whether it's by how you communicate your guess or how long you wait or something like that. Because then I would have to eat the whole cake all by myself...”
He grievingly glances down at his round Santa belly and asks: “Do you have any further questions?”
“Are the hats chosen at random?” Immanuel asks.
“Yes indeed, I will choose the colour of each hat completely at random.”
After some thought, the elves come up with a strategy that will guarantee them as much cake as possible. What is the maximum number of elves who are guaranteed to guess their hat colour correctly?
Possible answers:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
MY ANSWER:
For each person who gets told their hat color, someone has to waste their opportunity to guess the color of their own hat for someone else.
For example: There are 10 elves in total. The first elf doesn't know their hat color, so he shouts the tenth elf's hat color. The same goes for elves 2 and 9, 3 and 8, 4 and 7, 5 and 6. This means elves six to ten are guaranteed to guess their hat color correctly, whereas it's not guaranteed whether elves one to five will be able to guess theirs correctly. This means the maximum number of the elves who are guaranteed to guess their hat color is 5.
So I've asked this in a different forum as well and the answer is probably over 5. Could someone tell me how to solve this and give me an explanation as to why it is the right answer? It would mean a lot.
Thank you for your time.