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 :
...
- 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, ...
- 827
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 ...
- 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 ...
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33
votes
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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:
- 69.3k
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 ...
28
votes
Four prisoners wearing black and white hats
There are only 6 possible configuration of hats.
wwbb
wbwb
bwwb
...
- 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 ...
- 14.3k
23
votes
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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.
...
- 5,846
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
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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
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 ...
- 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?
- 420
17
votes
Accepted
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 ...
- 45k
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, ...
- 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 ...
- 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 ...
- 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 ...
- 44.8k
16
votes
Accepted
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.
- 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)...
- 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.
...
- 114k
15
votes
Accepted
Who will find the number on their own hat first?
Let's start with some ASCII art as a reference:
...
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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 $z=2$.
...
- 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 ...
- 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 ...
- 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):
- 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 ...
- 31.5k
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