Your plane has crash landed on a remote island called Logic Island. The local people have taken you and the rest of the people on the plane hostage. These local people are lovers of all things logic and have devised a game for you to test your logic in order to escape from Logic Island. If you fail, they will eat you and your group however if you score 90% of more, they will let your whole group go and transport you by raft to the mainland where you will reach safety.
The Logic Island leader explains the game to your group.
“We will align all twenty people of your group in a straight line so you are all facing west. The person at the back, number 20, will be able to see 19 people in front of them, the person in front of number 20, number 19, will be able to see 18 people in front of them….right up until the person at the front, number 1, who will have no-one in front of them. We will then place red or blue hats on every person starting with person number 20. Each individual person will not be able to see the colour of their hat or the hat colour of the people behind them. Each person will only be able to see the hat colours of those in front of them.
The arrows indicate the way each person is facing.
<–1 <–2…… <–18 <–19 <–20
After all the hats have been placed on heads, I will ask person number 20 what colour their hat is. I will then ask person 19, then 18 etc all the way to person number 1. You will only be allowed to reply “red” or “blue” in a monotone voice. As a group, you can give any meaning to the words “red” and “blue” however when we ask you what colour your hat is, we will only accept you saying “red or “blue,” If you say any other words or do not comply with using the monotone voice, we will eat your entire group. If we find that you use any type of physical code, look behind you, or take the hat off of your head to see its colour, we will eat your entire group. Once person 20 makes their guess, we will say “correct” or “incorrect.” Once person 19 makes their guess, we will say “correct” or “incorrect”, this will continue right up to and including person number 1.
You have 1 hour, as a group, to devise a strategy to maximise the correct ‘colour of hat’ guesses. Remember, if as a group, you get less than 90% of your guesses right, we will eat all of you. Note that we will be placing skull caps on your heads and you will not be able to see the colour of your own hat or those behind you, only those in front of you. Your time to plan a strategy as a group begins now.”
Clue (highlight to read): It is possible for at least 95% of your group to correctly state their hat colour without taking a wild guess. Remember that each person in your group can see the hat colours of those in front of them. Also each person in the group will be able to hear whether those behind them have correctly guessed. It is key to give different meanings to the words “red” and “blue” than colours.
Answer (highlight to read): The trick to this logic puzzle is to assign a different meaning to the words “red” and “blue.” Let us give the word “red” the meaning “odd number” and the word “blue” the meaning “even number” for person number 20. When person 20 is asked what colour their hat is, they will count how many people are wearing red hats in front of them. If this number is odd, they will say the word “red”, if this number is even, they will say the word “blue.” The logic people leader will then say “correct” or “incorrect.” The other 19 people in the line will be able to hear everything that has been said.
Let us imagine that 11 people in front of person 20 are wearing blue hats and 8 wearing red hats. This can be the random sequence:
R B B B R R B R B B R B B R B R R B B R
<– person 1 person 20 –>
Person 20 will say “blue” when asked what colour their hat is, as blue is code for even number of red hats in front of person 20. The Logic Island leader will then say “incorrect.” Person 19 will therefore know that there are an even number of red hats in front of person 20. They will also know that there are even number of red hats in front of them (from looking ahead) therefore person 19 must be wearing a blue hat. They say “blue” and the Logic Island leader says “correct.” Person 18 knows that there are an even number of red hats in front of person 19 however an odd number of red hats in front of person of them, therefore they say “blue.” This continues right up until person number 1. This results in a success rate of 95% and freedom from the Logic Island people.