70
votes
Accepted
Guessing hat colors. 4 prisoners
This is a bit of a stretch and I'm not sure it'd even work, but...
- 1,478
33
votes
Accepted
6 prisoners, 2 colors, one mute
It will be
Both A and B can see, what C sees, and that's why they both know that
From there, the problem reverts to the earlier one:
- 69.2k
24
votes
Guessing hat colors. 4 prisoners
A simple way would be that
Either there is, or there isn’t, but since everybody guessed the same, everyone will be right, or everybody will be wrong. No-one can be wrong by more than one, so whenever ...
- 69.2k
24
votes
Infinitely many hats
Here is another strategy, which requires looking at just 1 other person's hat:
- 13.5k
23
votes
Accepted
Cooperative guessing game: no incorrect guesses
The players can achieve $15/16$ win probability, which is optimal.
Associate the 15 players with nonzero vectors in $\mathbb{Z}_2^4$ like $(0,1,1,1)$. Let $S$ be the sum (entry-wise XOR, or Nim sum) ...
- 23.7k
22
votes
Accepted
21
votes
Accepted
3 Numbers on Hats, A = B + C
Answer:
First, some observations
Alright. Now, let's take a look at different possibilities:
Generalizing this, we get the rule above:
Do note however that the actual answer is actually slightly ...
- 4,607
19
votes
Accepted
Alternating Hat Colors
Before the "game", the prisoners decide on this strategy:
Assign an integer 1, 2, ..., n to each prisoner arbitrarily and call this the "prisoner number". When the numbers are shown, each prisoner ...
- 6,524
19
votes
Accepted
18
votes
6 prisoners, 2 colors, one mute
I'll try another explanation(with same result):
Here are the steps:
Next step:
So: Who talks?
- 420
18
votes
Accepted
16
votes
Five hats and four logicians in a circle
First observations
Since the puzzle is symmetric in white and black hats, I assume that the first man has a black hat. Then only three cases remain: (1) BWWB; (2) BWBW; (3) BBWW.
($\ast$) If a man ...
- 44.8k
14
votes
Accepted
Hat Guessing Game in Vegas
The players can win with probability
This is optimal because:
A strategy that achieves this is:
- 23.7k
13
votes
Accepted
100 Dwarves and a tiny room
No such algorithm exists.
Suppose Bob sees 0's on 99 shirts. He must call out 0, since it is possible that 0 is the only represented number.
Suppose Bob sees 0's on 98 shirts, on everyone except ...
- 31.4k
13
votes
Accepted
100 Dwarves vs. the Evil Elf
I will describe a lower bound and an upper bound on the probability of success the dwarves can guarantee. These bounds are:
Lower bound: Imagine that there is a stack of 100 shirts in the corner of ...
- 13.9k
13
votes
13
votes
Accepted
13
votes
Sorting kids by hat colours
I think this is a classic (and probably a duplicate):
To separate the two groups cleanly
- 2,672
12
votes
Infinite wizards and hats
Create infinite groups of finite size with $$size = (2^k)-1 ; k=2,3,...,infinity$$
Each of the groups will handle themselves as per the optimal strategy in this answer by xnor
Thus each group has a ...
- 221
11
votes
The sadistic executioner (a.k.a the 100 prisoners)
This is a fantastic puzzle, one of the best I have had to solve, so give yourself time to think it over. It took me a month to come to the solution.
To make you guys understand the solution, I will ...
- 295
11
votes
Accepted
Five hats and four logicians in a circle
Call the logicians A, B, C, and D, in the order they speak. WLOG, say A is wearing a white hat. Note that each logician sees 2 black, 1 white if he's wearing white or 2 white, 1 black if he's wearing ...
- 114k
11
votes
Accepted
Surviving Captain Nefarious
This is a proof that it is impossible to beat $\frac{7}{9}$.
There are 27 equally likely situations for the choice of colors. Each person will guess correctly in 9 situations. However, for any two ...
- 33.3k
11
votes
Accepted
A hat puzzle involving wizards
One possibility is
It is not possible for the wizards to do better. There are $16$ possible combinations of hat colors. Of these, $7$ of them have at least one wizard with no black hat, so cannot be ...
- 13.9k
11
votes
Numbered hats, Warden and Maths
You're all making this much more complicated than it needs to be. :)
Let's take an example:
- 5,621
11
votes
Accepted
Hat Puzzle with 5 different colours and 3 people
I think they can get all the way to
probability of getting all three guesses right.
There's probably a possible explanation that utilises Galois fields, modular exponents and discrete logarithms, (...
- 69.2k
10
votes
Accepted
MORE Prisoner Hats!
If the PhDs were allowed to plan a strategy together ahead of time, then the following would work:
Unfortunately, the king forbids them from collectively planning. Each couple may come up with a ...
- 31.4k
10
votes
Accepted
Guess simultaneously, color of your hat, no passing allowed
Here is a way to guarantee 500 correct guesses.
Edit:
The above is optimal, because the expected value of random guessing is 500, and with no extra information given to the logicians about how the ...
- 44.9k
10
votes
Infinitely many hats
So I just wanted to solve an amazing extension of this puzzle that I happen to know of.
Suppose that there were more than two colors. In fact, let's suppose that there were an uncountably infinite ...
- 9,722
9
votes
Accepted
N logicians wearing hats of N colors
There are two secrets to this question. First is to realise that you want no player in the circle to guess that the circle looks the same. If they guess the same circle, it is a wasted guess.
For ...
- 9,002
9
votes
100 Dwarves and a tiny room
I think f'' basically has the right answer, but here's a simpler explanation of why it's not possible.
Assume there is an algorithm. Consequently there must be an output i when all the shirts have ...
- 683
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