Remember the puzzle I posted last Saturday? https://www.rajibroy.com/?p=18817 The real problem in the book “The Riddler” (thank you Matt Moore for giving me that book) had 7 in the team – not 3. And the answer the book has 7/8. I am getting a far better probability. What am I doing wrong?
The problem, to remind ourselves is – randomly a black or white hat will be put on each of 7 of us. We cannot see our own color but we can see everybody else’s color. When called upon to guess our own color, we can call Black or White or we can say Pass. If one calls Black or White and is right, then the whole team wins. If it is wrong, then the whole team loses. However, if Pass is called, another person in random is asked to guess the color on their head. If everybody Passes then the whole team loses. What is the strategy to maximize the chance to win and what is that probability?
I am attaching the answer in the book which comes to 7/8 (using XOR logic).
My answer is the following:
The first person : If he sees the same color on the other six, he randomly calls a number. Else he says Pass.
The second person : Now he gets a chance only if the first person called Pass. Which means the first person has effectively said “All of you DO NOT have the same color”. So, if the second person sees the same color on everyone of the rest five, he simply calls the opposite color. He is guaranteed to be right. If he sees NOT all hats with the rest five to be of the same color, he simply says “Pass”.
The third person : uses the same logic. If the rest four have the same color, simply call the opposite color. Else say Pass.
This will go recursively and somebody is guaranteed to get the right answer. (In the absolute extreme case, The fifth person will call Pass – which is telling the sixth and seventh person – “Hey you two have different colors” and the sixth person can see the seventh person’s hat color….)
So independent of how many original players were there, there are only 2 cases the team loses – when the first person saw everybody else having the same color (all white or all black) and his random guess of his own color turned out to be wrong.
So, they lose with a probability of 2 / (2 the power n)
Winning probability is (1 minus the above) – which is much higher than 7/8
What am I doing wrong?
It was the second day of the conference I was attending in Orlando. I knew I was in a conference since my pedometer was clocking 20,000 steps every day. (Well, either that or New York city).
In any case, I was milling thru the booths in the exhibition floor talking to the companies that had come there when I suddenly heard my name being yelled out from behind. Now, as a background, I joined a new job six months back and this is a completely new industry for me. There is nobody from my past that I can think of that is now in the same industry that I would expect to run into in a conference.
But there was Arthur Altman! Thanks to Facebook, he had recognized me straightaway. Then again, there was not another shaved head Indian in the whole conference either.
Fancy meeting Arthur in Orlando! We got to know each other i2! That was in 1995 in Dallas. We worked together for a few years – even had a common boss for some time and then split in the early 2000s. The last time I saw Arthur was about 15 years back or so.
After that, it has been those annual chats during my birthday calls. And then finally, yesterday I ran into him!!
We caught up over drinks about our old company, some of our old friends from work as well as his family.
Arthur still is – as he always was – bubbling with energy, full of ideas and always quick on the draw. It is like nothing has changed at all. Well, the beard has decidedly grown longer. There is always that 🙂