Puzzle 119 - solution

They decide that everyone will say white if they see odd number of white hats in front of them. If they see even number of white hats in front of them, they say black.

Let us see how this works.

• The last person sees ahead and let us say he sees odd number of white hats and he says "white"
• The person in front of him sees ahead. And if he sees odd number of white hats in front of him, he realizes his hat must be black and he says "black". Instead if he sees even number of white hats in front of him, and since the person behind him had seen odd number of white hats, he realizes his hat is white and says "white".
• The third person also repeats the same procedure. 
• This is repeated until you reach the last person. If the person behind him says 'white', his hat is white and he says 'white'. If not he says 'black'.

Let us see another possibility

• The last person sees even number of white hats and says 'black'
• If the second person sees even number of white hats, he says 'black'. Instead if he sees odd number of white hats, he says 'white'.
• And so on.  

This way each person has guessed the color of his hat correctly. Except for the last prisoner. And he has 50/50 chance of guessing correct color. 

So everyone else is saved and last person has 50/50 chance of survival.  

The solution to this puzzle is explained in this TED-ed video.