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“I don't know the numbers”: a math puzzle

alexanderell.is

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Re: “I don't know the numbers”: a math puzzle

#91

Post by vintermann »

refurb wrote: post #84
seoaeu wrote: post #47

Did I miss it, or does the article not actually explain why we know that at least one pair can be eliminated every round? It seems plausible that the process could get "stuck" in a place where Peter and Sandy both don't learn anything new from discovering the other doesn't know the answer yet. (The puzzle being solvable means that they can't actually get stuck, but that feels like outside information...)

Yup, without a critical assumption - each "I don't know eliminates a potential solution" - it's not solvable because there is no new information when someone says "I don't know the answer"

For N=100, though, they could know beforehand that potential solutions are eliminated at least the first 15 times someone says "I don't know". For any N, it should be possible to work out the longest such a game could go on (until one of them says, "I don't know, and I don't know any more from your last statement").

Re: “I don't know the numbers”: a math puzzle

#92

Post by vintermann »

rexpan wrote: post #82

My SQL Solution (SQL Server) -- Create Table #x(v) from 1..99 WITH x AS (SELECT n FROM (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9)) v(n)) SELECT ROW_NUMBER() OVER (ORDER BY (SELECT NULL)) AS v INTO #x FROM x ones, x tens ORDER BY 1 ; DELETE FROM #x WHERE v > 99 ; -- Create Candidate #c(x, y, s, p) with pair (x,y) (x

It's not pretty, but it works!

Re: “I don't know the numbers”: a math puzzle

#93

Post by tobr »

hvdijk wrote: post #90
tobr wrote: post #89

I understand it’s a puzzle and it requires some suspension of disbelief, but I think Peter is wrong. He doesn’t know the numbers after 14 tries. He is assuming that Sandy has figured out the trick and is able to correctly keep track of 9801 + 198 = 9999 lists of number pairs in her head. As far as I can tell by how the puzzle is phrased, that’s a totally unreasonable assumption. It’s more likely that she says “I don’…

We are given that he says he does know and have no reason to doubt it. It is possible that Peter and Sandy know one another and already know that they are perfect at figuring out maths/logic puzzles.

I would phrase the puzzle differently.

They both need to say they know, otherwise we have no reason to believe Sandy has figured it out.

And they both need to be right, which we also have no reason to believe at the moment.

Re: “I don't know the numbers”: a math puzzle

#94

Post by Sankozi »

lisper wrote: post #3

Bonus meta-puzzle: the puzzle as stated is actually unsolvable. Both Sandy and Peter have to know something that the puzzle implies but does not actually stipulate that they know. What is it?

Each one of them:

1. is a person that solves logical puzzles if possible

2. knows that the other one + (insert point 1 here).

3. knows that the other one + (2)

4. knows that the other one + (3)

...

Re: “I don't know the numbers”: a math puzzle

#95

Post by hvdijk »

tobr wrote: post #93
hvdijk wrote: post #90

Earlier quoted context omitted.

We are given that he says he does know and have no reason to doubt it. It is possible that Peter and Sandy know one another and already know that they are perfect at figuring out maths/logic puzzles.

I would phrase the puzzle differently. They both need to say they know, otherwise we have no reason to believe Sandy has figured it out. And they both need to be right, which we also have no reason to believe at the moment.

The puzzle doesn't state whether Sandy has figured it out because that information is not necessary to figure out the solution. We can infer that she has (under that implicit assumption that she is a perfect logician) by the fact that we know less than she does and still have enough information, though. As for that last part: we also don't truly know that Peter isn't lying just to mess with the puzzle, I guess. That is another implicit assumption.

Re: “I don't know the numbers”: a math puzzle

#97

Post by drexlspivey »

Someone wrote: post #56
Arainach wrote: post #30

This isn't quite right. This does find the solution when the problem is well-formed, but it doesn't prove that the solution is valid: if round == 15: for product in products: if len(products[product]) == 1: print('Peter: I do know the numbers') print(products[product][0]) return This should not return - it should continue iterating. If the problem (and the code) is correct, then it should only find one answer - but u…

The ‘normal’ way to write something like that (I hope I got the range ends right) is for i in range(1, N): for j in range(1, j + 1): candidate_pairs.add((i, j)) Also, conceptually, you don’t have tuples, but multisets. If you were to use those, you wouldn’t need that if at all. Multiset isn’t built-in to python, so you’d need to use an external package. Alternatively, write a class for storing pairs of ints i , j tha…

Python actually has built in support for multisets, you use a set of frozensets

>>> a = set( (frozenset((1,2)), frozenset((3,4)) ) )

>>> frozenset((2,1)) in a

True

Re: “I don't know the numbers”: a math puzzle

#98

Post by drexlspivey »

Sankozi wrote: post #94
lisper wrote: post #3

Bonus meta-puzzle: the puzzle as stated is actually unsolvable. Both Sandy and Peter have to know something that the puzzle implies but does not actually stipulate that they know. What is it?

Each one of them: 1. is a person that solves logical puzzles if possible 2. knows that the other one + (insert point 1 here). 3. knows that the other one + (2) 4. knows that the other one + (3) ...

This is called Common Knowledge in game theory:

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum. It can be denoted as {\displaystyle C_{G}p}.

Re: “I don't know the numbers”: a math puzzle

#99

Post by refurb »

vintermann wrote: post #91
refurb wrote: post #84

Earlier quoted context omitted.

Yup, without a critical assumption - each "I don't know eliminates a potential solution" - it's not solvable because there is no new information when someone says "I don't know the answer"

For N=100, though, they could know beforehand that potential solutions are eliminated at least the first 15 times someone says "I don't know". For any N, it should be possible to work out the longest such a game could go on (until one of them says, "I don't know, and I don't know any more from your last statement").

I would assume it hinges on the fact that when one party says "I don't know the answer" then they are eliminating a possibility (which isn't intuitive at all with that answer).

Re: “I don't know the numbers”: a math puzzle

#100

Post by vintermann »

refurb wrote: post #99
vintermann wrote: post #91

Earlier quoted context omitted.

For N=100, though, they could know beforehand that potential solutions are eliminated at least the first 15 times someone says "I don't know". For any N, it should be possible to work out the longest such a game could go on (until one of them says, "I don't know, and I don't know any more from your last statement").

I would assume it hinges on the fact that when one party says "I don't know the answer" then they are eliminating a possibility (which isn't intuitive at all with that answer).

I believe (I could be wrong here, I haven't rigorously checked this) that when the numbers are less than 100 and both parties know this, there will always be a possibility eliminated by a "I don't know" statement.
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