A nasty logic teacher tells the students the following durin

A nasty logic teacher tells the students the following: during the next class everyone will line up facing the front of the line, and the teacher will place a hat on each student’s head. The hats are either red or white and each student can only see the hats of the students in front of them, not their own or the ones behind. Starting at the back of the line the teacher asks the students one by one “What is the colour of your hat? If the answer is correct the teacher says so and the student gets an A in the course. If the answer is wrong, the teacher says “Wrong, you fail.” All students hear all the answers and whether they are right or wrong.

There are n students in the class and they have a couple of days to come up with a plan to get as many As as possible. They are very bright, and manage to come up with something that works almost 100%, regardless of the distribution of colours of the hats and without any cheating (peeking, using tone of voice, secret text messages, etc.). What was their strategy?

Solution

Solution : The strategy can be decided after keeping two important things in mind or prior knowledge of the two facts (i) The teacher start from the back of the line and (ii) All students hear all the answers and whether they are right or wrong.

Since the teacher starts asking from the student at the back, so the very first student ( the one at the back) has 50% chance of failing as he cannot see the color of his own hat, but he can see the colour of the hat of the students in front of him (he will act as the key to start of the strategy).

Strategy used can be described as follow:

The first student will reply instantly, if he see a white hat on the next student and he delayed his response by say 1 min (the time can be decided by the mutual undertsnding of all the students), if he see a red hat on the student just next to him. Now depending on the time taken for the response by the first student the next student can get the idea of colour of his hat ( \"White\" if the first student has given instant response and \"Red\" if the first student has taken delayed time or the pre-decided time say 1 min)

Now the same strategy will be used by every student to save the next student ( they will reply instantly if the hat on the next student is \"White\" or they delay their esponse by 1 min if the hat on the next student is \"Red\")

Using this strategy it is certain that (n-1) of the n students will get an A but the chances for the first student to get an A is only 50% because he do not have a preceeding student, who can use the same strategy for him. So we can say he is at risk but he is playing the most important role to initiate the chain of securing an A for other students. He may get an A if his response ( obiviously a guessed response) proves out right. In that case we say all n students (including him) will get an A.

Using the above strategy, students can make sure that at most one student may fail the exam and other n-1 students will get A or there may be possiblity of getting all students getting an A.