Question

The wizards have an infinite supply of white and black hats. They place either a white or black hat on the head of each dwarf. Then, starting with the tallest dwarf (in the back of the line), they ask each what color hat he is wearing. If the dwarf answers incorrectly, the wizards kill him (the other dwarves can hear his answer, but cant tell if he was killed or not). What is the most number of dwarves that will be killed using this optimal strategy?

Answer 1

Answer:

1

Step-by-step explanation:

Answer is 1 Since the dwarves beforehand made a strategy. The strategy goes like this:-

The tallest dwarf(last in line) will count the no. of white hats in front of him i.e(since he can see all hats except his own) and will say his hat is white if the no. of white hats in front of him are even and will say black if the no. of white hats are odd.

So,the tallest dwarf has the prob. of living=50%

Depending on the answer of tallest dwarf everyone would get to know about the total white hats are even or odd. let us suppose he said ‘WHITE’ for the sake of simplicity i.e total EVEN no. of white hats.

Now for the second tallest dwarf he will count the no. of white hats in front of him:-

if it is odd, but according to answer of tallest dwarf total white hats are even so it means that he himself is wearing a white hat, so he will answer white.

2. if it is even, but according to answer of tallest dwarf it is also even this means that he is wearing a black hat, so he will answer black.

Now Similarly everyone will keep track of no. of times “WHITE” have been spoken before their own turn (excluding the first time since it was just to tell everyone that total white hats are even).

And on the basis of that they get to know whether the total white hats are even or odd including their own hat.(like suppose white has been spoken 3 times excluding first time then now total no. of white hats are odd ).

Now they count the no. of white hats in front of them and compare these two like i have explained previously to determine the color of their own hat.

Hence everyone will be able to tell the color of their hat except the tallest dwarf if who is lucky enough(50%) will live else we lose only 1 dwarf.