A convention is held where all the big logicians are summoned. The master places a band on everyone’s forehead. Now all of them can see others’ bands but can’t see his own. Then they are told that there are different colors of bands. All the logicians sit in circle and they are further explained that a bell will ring at regular intervals. The moment when a logician knew the color of band on his forehead, he will leave at the next bell. If anyone leaves at the wrong bell, he will be disqualified.
The master assures the logicians that the puzzle will not be impossible for anyone of them. How will the logicians manage ?
First the logicians will have to take a leap of logic. The master has told them that it won’t be impossible for any logician to solve the puzzle. Thus it is assure that any of the color can’t exist only once. If it did, the one wearing it will have no clue about that color which would be unfair for him.
Now every logician will look around in the circle and count the number of times they see a particular color. If a color is seen only once, then the logician will know that the color on his band must be of the same color (as per the leap of logic). And then the logician will leave on the first bell.
In the similar fashion, any logician who see any color just once will be able to identify his own color and they will leave when the bell rings or they will be disqualified and asked to leave. Equivalently, any color for which there are two bands, will be eliminated after the first bell has rung. Thus there must be three bands of any remaining color at least.
Assume that a logician don’t see any color once, but sees a color twice. If they were the only bands of this color then the two logicians must have left at the first bell. But they did not. Thus it means that his band color is the same and he will leave on the second bell.