Imagine you and nine other people have been captured by super-smart aliens. These aliens think humans might be delicious, but their rules say they can’t eat beings that are logical and work well together. To see if you qualify, they give you a tricky test.
The alien guarding you explains the test through a universal translator. You will all stand in a single-file line, facing forward, arranged by height. This means each person can see the hats of everyone in front of them but not their own or those behind them. Each of you will wear a hat that is either black or white, chosen randomly. The alien won’t tell you how many hats of each color there are.
When the test begins, starting with the person at the back of the line, each of you must guess the color of your own hat. You can only say “black” or “white,” and you can’t use any other signals. If at least nine of you guess correctly, you’ll all be set free. You have five minutes to come up with a plan before the test begins. Can you think of a strategy that will save everyone?
The key to solving this puzzle is using the first person’s guess to communicate important information. The person at the back of the line can see all the hats in front of them. They can use their guess to tell everyone else something important about the hats they see.
Instead of trying to count the exact number of black or white hats, the first person can focus on whether the number of black hats they see is odd or even. This is called “parity.” The plan is for the first person to say “black” if they see an odd number of black hats and “white” if they see an even number.
Let’s say the hats are arranged like this: the tallest person sees three black hats in front of them. They say “black,” indicating an odd number of black hats. They might guess their own hat color wrong, but that’s okay because only one wrong guess is allowed.
The second person sees an odd number of black hats, so they know their hat must be white and guess correctly. The third person sees an even number of black hats, so they know their hat is black. This pattern continues, with each person using the information from the person behind them to figure out their own hat color.
By the time it reaches the person at the front of the line, they can use the information from the ninth person to determine their hat color. This strategy works no matter how the hats are arranged. The first person has a 50% chance of guessing their own hat color wrong, but their guess helps everyone else get theirs right.
So, thanks to this clever plan, you and your friends are safe, and the aliens will have to find another meal. This riddle shows how teamwork and logical thinking can solve even the trickiest problems!
Gather your classmates and role-play the prisoner hat riddle. Assign roles and use colored paper hats to simulate the scenario. Practice the strategy of using parity to guess the hat colors. This will help you understand the logic behind the solution and improve your teamwork skills.
Create your own puzzles using the concept of parity. Design scenarios where you have to determine the parity of a set of objects. Share your puzzles with classmates and solve them together. This activity will reinforce your understanding of parity and logical reasoning.
Write a short story where characters face a challenge similar to the prisoner hat riddle. Use logical thinking and teamwork to solve the problem in your story. Share your story with the class and discuss the different strategies used. This will enhance your creative thinking and application of logic.
Play a game where you randomly assign black and white hats to classmates. Each student must guess their hat color based on the parity strategy. Keep track of correct guesses and discuss the probability of success. This game will help you understand probability and improve your strategic thinking.
Organize a workshop where you and your classmates brainstorm different strategies for solving logical riddles. Discuss how teamwork and communication can improve problem-solving skills. Present your strategies to the class and receive feedback. This workshop will enhance your collaborative and critical thinking abilities.
You and nine other individuals have been captured by super intelligent alien overlords. The aliens think humans look quite tasty, but their civilization forbids eating highly logical and cooperative beings. Unfortunately, they’re not sure whether you qualify, so they decide to give you all a test.
Through its universal translator, the alien guarding you tells you the following: You will be placed in a single-file line facing forward in size order so that each of you can see everyone lined up ahead of you. You will not be able to look behind you or step out of line. Each of you will have either a black or a white hat on your head assigned randomly, and I won’t tell you how many of each color there are. When I say to begin, each of you must guess the color of your hat starting with the person in the back and moving up the line. And don’t even try saying words other than black or white or signaling some other way; you’ll all be eaten immediately. If at least nine of you guess correctly, you’ll all be spared. You have five minutes to discuss and come up with a plan, and then I’ll line you up, assign your hats, and we’ll begin. Can you think of a strategy guaranteed to save everyone? Pause the video now to figure it out for yourself.
The key is that the person at the back of the line, who can see everyone else’s hats, can use the words “black” or “white” to communicate some coded information. So what meaning can be assigned to those words that will allow everyone else to deduce their hat colors? It can’t be the total number of black or white hats. There are more than two possible values, but what does have two possible values is that number’s parity, that is whether it’s odd or even. So the solution is to agree that whoever goes first will, for example, say “black” if he sees an odd number of black hats and “white” if he sees an even number of black hats.
Let’s see how it would play out if the hats were distributed like this. The tallest captive sees three black hats in front of him, so he says “black,” telling everyone else he sees an odd number of black hats. He gets his own hat color wrong, but that’s okay since you’re collectively allowed to have one wrong answer. The second prisoner also sees an odd number of black hats, so she knows hers is white and answers correctly. The third prisoner sees an even number of black hats, so he knows that his must be one of the black hats the first two prisoners saw. The fourth prisoner hears that and knows that she should be looking for an even number of black hats since one was behind her. But she only sees one, so she deduces that her hat is also black. Prisoners five through nine are each looking for an odd number of black hats, which they see, so they figure out that their hats are white.
Now it all comes down to you at the front of the line. If the ninth prisoner saw an odd number of black hats, that can only mean one thing. You’ll find that this strategy works for any possible arrangement of the hats. The first prisoner has a 50% chance of giving a wrong answer about his own hat, but the parity information he conveys allows everyone else to guess theirs with absolute certainty. Each begins by expecting to see an odd or even number of hats of the specified color. If what they count doesn’t match, that means their own hat is that color. And every time this happens, the next person in line will switch the parity they expect to see.
So that’s it, you’re free to go. It looks like these aliens will have to go hungry or find some less logical organisms to abduct.
Riddle – A puzzling question or problem that requires thought to solve. – The math teacher gave us a riddle to solve that involved finding the missing number in a sequence.
Strategy – A plan of action designed to achieve a specific goal. – Our strategy for solving the complex equation was to break it down into smaller, more manageable parts.
Guess – An estimate or assumption made without sufficient information. – When faced with a difficult math problem, sometimes making an educated guess can help you find the right path to the solution.
Hats – Objects often used in logic puzzles to represent different categories or groups. – In the classic logic puzzle, each person wore a different colored hat, and we had to figure out who wore which color.
Black – A color often used in logic puzzles to represent one of the possible options or categories. – In the puzzle, three people wore black hats, and two wore white hats, and we had to determine who wore which color based on the clues given.
White – A color often used in logic puzzles to represent one of the possible options or categories. – The challenge was to deduce who was wearing the white hat based on the logical statements provided.
Logical – Relating to clear, sound reasoning. – Using logical thinking, we were able to solve the problem by following each step carefully and ensuring our reasoning was correct.
Parity – The concept of whether a number is even or odd. – Understanding parity helped us solve the math puzzle, as we needed to determine if the sum of the numbers was even or odd.
Teamwork – The combined effort of a group to achieve a common goal. – Through teamwork, our group was able to solve the challenging math problem faster than if we had worked individually.
Problem-solving – The process of finding solutions to difficult or complex issues. – Problem-solving skills are essential in mathematics, as they help us tackle and overcome challenging questions.
