The person behind you gives you the parity and you deduce that you have a white hat. You then you add the hats in front of you, and they’re odd. The guard asks you your hat color. You say white, you go free. You say black, the person in front of you does. The answer should be 50, correct?

The answer, as you’ve stated the question, is 100. If you want the answer to be 99, then you must add “with certainty” to the explanation. Otherwise, there are two scenarios: one where 99 go free and one where 100 go free. The maximum of 99 and 100 is 100. Therefore, the answer is 100.

@alee Let’s say P99 is the only prisoner to wear a black hat.
P100 sees 1 black hat indicating an odd state(1). P99 looks forward and sees an even state(0). P99 does the math (P100)1- (forward)0 = (P99)1. Now he knows his hat is 1 or odd or black.

Now P98 sees even(0) knows P100 said black(1) and P99 said black(odd). P98 does the math. (P100)1 – (P99)1 – (forward)0 = (P98)0

isn’t the answer based on the assumption that there is an even number of black hats? for simplicity, imagine 99 prisoners had white hats and 1 guy had a black hat. the guy in the back of the line sees a black hat on someone, so his sum is odd, thus leading him to answer black which would be wrong.

100.

Tell each prisoner to look at their hat before answering.