57 votes
Accepted

In the 100 blue eyes problem - why is the oracle necessary?

Let's continue the induction, since the jump to 99 blue eyes does seem weird. After all, everyone knows that someone has blue eyes. If there are 4 blue eyed-people, A will look at B,C,D, thinking : ...
user avatar
  • 1,086
56 votes

In the 100 blue eyes problem - why is the oracle necessary?

Every blue-eyed person sees 99 blue-eyed people. Since they don't know that they have blue eyes, they suspect it might be the case that every other blue-eyed person can only see 98 blue-eyed people, ...
user avatar
46 votes
Accepted

Are there eighteen or twenty bars in my castle?

I think I have a faster solution than Rubio. (Or I did, when Rubio's solution took one day longer than mine; he's since incorporated my solution into his answer). The answer: The explanation: Let's ...
user avatar
  • 576
39 votes

Are there eighteen or twenty bars in my castle?

The answer Assumptions We assume both have at least one bar on their window (or the window couldn't be said to be barred, and they're told their windows are the only barred ones). We further assume ...
user avatar
  • 40.2k
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:
user avatar
  • 69.3k
29 votes

In the 100 blue eyes problem - why is the oracle necessary?

The only explanation I've seen that's sufficiently precise to be satisfying is this answer to the corresponding question on math.SE. The key fact that the "oracle" (guru) gives you, that you didn't ...
user avatar
28 votes

Four prisoners wearing black and white hats

There are only 6 possible configuration of hats. wwbb wbwb bwwb ...
user avatar
  • 9,012
26 votes
Accepted

A game of multiplication and addition

The numbers are Alice: You can't know my sum. Bob: Thanks to you, I know your sum. Alice: Then I know your product. EDIT: While I show the consistency of the two numbers with the provided ...
user avatar
  • 14.3k
23 votes
Accepted

Abby and Bobby and three numbers on the blackboard

The following table gives the numbers of one person and the corresponding possible numbers of the other person. ...
user avatar
21 votes
Accepted

A 3-person hat puzzle (no, not that one) (no, not that one either!)

The crucial fact here (which I think makes the question kinda unfair since it's not exactly common knowledge) is that And we had better Now, So And in this case
user avatar
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 ...
user avatar
  • 4,607
19 votes

In the 100 blue eyes problem - why is the oracle necessary?

I think considering it backwards might actually be the easier way to understand it. A given blue-eyed person does not want to leave, so he hopes he has brown eyes and assumes he has brown eyes. He ...
user avatar
  • 291
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?
user avatar
  • 420
17 votes
Accepted

Again! 6 prisoners, 2 colors, one mute

Reasoning:
user avatar
17 votes

Three people wearing hats

This was trickier than it looked. I have a feeling that there should be a quicker way to find the answer than the list of cases I worked through. Note: I interpret "two of which are factors of the ...
user avatar
17 votes
Accepted

Three secret numbers and sum

I think this works, although it results in two possible solutions that I believe meet all the criteria... The first student says that he knows that the two other students have different numbers, ...
user avatar
  • 101k
16 votes

In the 100 blue eyes problem - why is the oracle necessary?

The colour of the guru's eyes is not relevant. The guru is allowed to speak about eyes and nobody else is. If any blue eyed person said "I can see someone with blue eyes" where everyone on the island ...
user avatar
  • 5,413
16 votes
Accepted

Three secret numbers (Variant 1 and Variant 2)

X's statement means that his number must be even (if it was odd, Y and Z would have numbers summing to an even number, so they could be the same). In the first case, Y's statement means that Y has an ...
user avatar
  • 176
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 ...
user avatar
  • 44.8k
16 votes
Accepted

What are the three numbers in this puzzle?

The Answer: My reasoning: How I got the numbers:
user avatar
16 votes
Accepted

Sum of secret numbers is 101

Alice: I know we have different numbers. Bob: Aha, I got it. I found all the numbers. Charlie: Me, too. I know our numbers, now. Alice: Alas, I still don't know.
user avatar
  • 9,135
15 votes
Accepted

Farm dimensions

In other words, we are looking for two positive integers $x$ and $y$ with $x\ge2$, $y\ge2$ and $4\le x+y\le40$ that fit with the conversation. Let us denote the sum $s=x+y$ and the product $p=xy$. (1)...
user avatar
  • 44.8k
15 votes
Accepted

Two students guessing positive integer

Let $a$ and $b$ be the integers thought of by A and B, and let $N$ and $n$ be the integers given by the teacher, where $N>n$. Assume for contradiction that both A and B keep on saying "no" forever. ...
user avatar
15 votes
Accepted

Who will find the number on their own hat first?

Let's start with some ASCII art as a reference: ...
user avatar
  • 17.9k
14 votes
Accepted

Another sum and product puzzle

The fact $3z\geq x+y+z>xyz$ implies that $3>xy$, so $x=1$ and $y=1$ or $x=1$ and $y=2$. In the former one, we get $2+z>z$. Otherwise we get $3+z>2z$, so $z<3$ and then in fact $z=2$. ...
user avatar
  • 4,113
14 votes
Accepted

Constructing interesting "Don't Know - Don't Know - Now I know!" type puzzles

The professor says: I have written on Alice's paper an even ordinal and on Bob's paper an odd ordinal. Who has the lesser ordinal? Then, after $\lambda$ iterations of the game, it is common ...
user avatar
  • 7,731
14 votes
Accepted

Yet Another What am I? Puzzle

You are 1) From my birth I want to rise But I shall fall, its no surprise I should display approval from afar But pedantic snobs is what you are. 2) I could produce a rise in rank But not if my ...
user avatar
  • 2,204
14 votes

Communicating Information about Cards

I'm no mathematician, but I think both Deusovi and Gareth can say: I'm not sure about making it into a proof, but by the extreme example: Basically (maybe/hopefully):
user avatar
  • 36.4k
14 votes

A game of multiplication and addition

Riley's answer proves that the numbers are consistent with the conversation. I will prove that these are the only possible numbers. After Alice's first statement, After Bob's statement: After ...
user avatar
  • 31.5k

Only top scored, non community-wiki answers of a minimum length are eligible