# Tag Info

Accepted

### Guessing hat colors. 4 prisoners

This is a bit of a stretch and I'm not sure it'd even work, but...
• 1,488
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:
• 70.8k

### 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 ...
• 70.8k

### Infinitely many hats

Here is another strategy, which requires looking at just 1 other person's hat:
• 13.5k
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) ...
• 24.1k
Accepted

### Numbered hats, Warden and Maths

Sure, it's perfectly doable.
• 142k
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 ...
• 4,617
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 ...
• 6,554
Accepted

### 10 prisoners and 10 lists of numbers

The strategy is: Why it works: For example:
• 33.8k

### 6 prisoners, 2 colors, one mute

I'll try another explanation(with same result): Here are the steps: Next step: So: Who talks?
• 420
Accepted

• 128k

### 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 ...
• 45k
Accepted

### Hat Guessing Game in Vegas

The players can win with probability This is optimal because: A strategy that achieves this is:
• 24.1k
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 ...
• 31.7k
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

### Numbered hats, Warden and Maths

Everybody walks out free and happy.
• 5,501
Accepted

Solution: Edit:
• 391

### Sorting kids by hat colours

I think this is a classic (and probably a duplicate): To separate the two groups cleanly
• 2,757

### 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 ...
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 ...
• 115k
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 ...
• 33.4k
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

### Numbered hats, Warden and Maths

You're all making this much more complicated than it needs to be. :) Let's take an example:
• 5,631
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, (...
• 70.8k
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 ...
• 31.7k
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 ...
• 46.6k

### 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 ...
• 9,792
Accepted

### Hat-guessing with a spy

To start: Then: Total losses: Example: Same example, no spy: Either way, the spy only affects the outcome for themselves and the person after them in the line This only works for N > ...
• 8,588
This solution gives a 7/9 probability of survival. As JonTheMon has explained, agreeing on a cycle of A $\to$ B $\to$ C $\to$ A is necessary to ensure that the same person won't be chosen twice. This ...