The Labyrinthians Exchange... Solved.... I Think :)

[HyToFry]

Extro... you wrote:

quote:

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Paul: I'm afraid I've forgotten the address. I can only remember the product of the two numbers, and that the first number wasn't greater than the second.

This just tells us Paul knows first*second, and that first<=second. So we eliminate all solutions with first>second.

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Now according to your calculations, which are correct, the numbers cannot = 4,13

4*13 has a product of 52 and 2,26 is the only other set of numbers that have this product...

NOW with 2,26 if paul knows that q<=p<=100, and the only numbers that have only one possible outcome that can be added up to get 28 are 5,23 and 11,17, and all other numbers that can add up to 28 have multilpe sumroots (they do i checked them) THEN then the prime roots would have to be 5,23 or 11,17, BUT because 5*23 = 115 and 1,115 is the other possibility in this scenerio, Sam would be able to rule out the possiblility of 5,25 being the answer, because if it was Paul would know that 5,25 is the answer (because he would have a sum of 115 and 1<=115<=100 is not a true statement and Paul would know this from the start, but he didn't and Sam knew that, so she could rule it out... GET IT?

The same is true for 11,17 although i'm not going to calculate it out for you.

So as far as Paul knows the answer could still be 2,26 - because all other roots of 28 have multiple answers that can be the answer, and thus wouldn't be able to tell if Sam thought the Sum was 28 or 17 as both would let sam know that all possible roots have products with multiple roots

So this goes back to my former statement, IF Sam and Paul DO know the size of the town the answer CANNOT be 4,13 and as you (extro) have pointed out this would result in 24 possibilities, However if they DON'T know the size of the town, 4,13 is the only possible answer.

this is why i have decided to go with 4,13 and its not because of ANYTHING ANYONE HAS SAID that changed my mind (with the exception of Extro... who showed me that when S and P know the size of the town, there are 24 possible outcomes).

Comments?

[This message has been edited by HyToFry (edited 03-10-2000).]

[araya]

Well, congratulations on finally getting the correct answer then, even if all the help everyone tried to give you apparently didn't amount to anything (LOL)..

The problem states that P and S are able to deduce the address of the party solely from the information they exchange in the dialog. It does not state that they know the limits of the city. If you assume that they know that the city is 100x100, then perhaps there are multiple answers, but that doesn't show the question is ambigious, it shows that your assumption is wrong.

[Ghost Post]

First, a clarification: You wrote
"when S and P know the size of the town, there are 24 possible outcomes"

I never considered the possibility that they don't know the size of the town. I got the 24 solutions (including 2,25, but not 4,13) when I assumed that the first two statements by Paul and Sam, that they can not remember the address, meant also that they could not figure it out. I assumed they knew the size of the town.

Now, I'm still having trouble parsing your explanation of what is wrong with 4,13.

Paul knows the product is 52, but can't figure out if the numbers are 4,13 or 2,26. (He DOES already know it isn't 1, 52, because Sam already said both numbers were greater than 1)

Sam knows the sum is 17, which could mean the numbers are any of (2,15) (3,14) (4,13) (5, 12) (6, 11) (7, 10) (8, 9). For any of these, the product can only be factored one way. So she knows from the sum that Paul didn't know the numbers.

On the other hand, if she had known the sum to be 28, she would have to consider that the numbers might be (5,23) or (11,17), in which case Paul WOULD have known the numbers from their product. 5*23 = 115 = 1*115, but Paul ALREADY KNEW that neither number was 1 (the size of the town would have also clued him in, but it is not essential here). So, if the sum were 28, Sam could not have known that Paul couldn't figure out the numbers from the product.

So, when Sam say's she knew he couldn't figure it out, that eliminates 28 as the sum, which eliminates 2,26, which leaves only 4,13 as possible numbers with the product 52.

I think I know what you did though. When Samantha says "I knew you couldn't figure out the address", the question is: WHEN did she know? I am assuming she meant she knew that prior to him stating it, but after he was told by her that neither number was a 1. In other words:

1) Paul say's he knows the product, but can't remember the numbers.
2) Sam say's she knows the sum, can't remember the numbers, but remembers neither was a 1.

At this point, Paul knows neither number was a 1, and at this point, Sam knows Paul can't (even with knowing that neither number was a 1) deduce the numbers, but Paul doesn't know she knows that.

3) Paul says he can't deduce the numbers.
4) Sam say's she knew that.
5) Paul - Now I do.
6) Sam - So do I.

[HyToFry]

Okay if P has a product of 52, then his possible root sums are 1,52 (which is eliminated by S in the 2'nd statement) 2,26 and 4,13

Consider S = 28

All of the products of the root sums of 28 have AT LEAST two possible productroots.

Consider 3,25
3*25 = 75, also 5,15 is 75
All numbers that add up to 28 are the same (do the math yourself)

The only exceptions are 5,23 and 11,17 Which P would Know that S would Know that if it was either of these numbers P would know the address from the begining, (Stateing that the problem never says that P only forgot the answer, and not that he couldn't figure it out is pretty far fetched) So because S would know S would have imediantly ruled these out, Due to the fact that P would know the answer in round one. With this in mind I have eliminated the other two possiblities and since all other possibles have multiple product roots, YOU GET THE PICTURE I'M SURE.

Now P would know that S would know this, so either way S=28 or S=17, S would know that P can't possibly be helped with the knowledge that niether number was 1, and so either way S's statement remains true...

When P said I still don't know
and S said I already knew that
P could have said at this point "But did you know that i knew you would know that?"
(because he did)


Comments?

[Ghost Post]

HyToFry wrote: Stateing that the problem (never?) says that P only forgot the answer, and not that he couldn't figure it out is pretty far fetched.

OK. If you take "I forgot" (as stated in problem) to mean "I forgot, and I can't figure it out from the information I have available", then you get the 24 possible solutions, one of which is 2,25. Any one of these 24 would allow Paul and Samantha to figure out the exact address from the conversation, although we, from overhearing the conversation, but knowing neither the sum nor product, can't tell exactly which house it is.

[HyToFry]

Array you said

quote:

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Well, congratulations on finally getting the correct answer then, even if all the help everyone tried to give you apparently didn't amount to anything (LOL)..
The problem states that P and S are able to deduce the address of the party solely from the information they exchange in the dialog. It does not state that they know the limits of the city. If you assume that they know that the city is 100x100, then perhaps there are multiple answers, but that doesn't show the question is ambigious, it shows that your assumption is wrong.

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If this is the case, then the whole riddle doesn't work... you assume things to with your deduction of 4,13
1. Your Assuming that P can add and S can multiply.
2. You assume that P and S know all of the Prime numbers.
It never states that P and S know all of the prime numbers (and they would have to to know the answer to the puzzle, or if they didn't then P would have to be able to add and S able to multiply).

It does say they are "quick witted" but i think if they were quick witted then P would DEFINATELY know that 1,115 could not be in the town, and S would know this (eliminating the possibility that it could be 3,25) So S still knows that knowing neither number was one would help P. and so P will determan that 2,26 is still possible.

Also we assume that they did figure out the answer at the end (it doesn't say that they figured it out, only that they now know the answer), if we can assume nothing, then how do we know that they didn't just SEE the house? in other words.. Some things HAVE to be assumed in order for the puzzle (in its current form) to be solved at all

More Comments?

[HyToFry]

Also, looking through the other topics, I noticed a flaw in your thinking as well:

Gidon Wrote

quote:

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Hi
I looked at the old thread aswell, and wrote a program to generate the numbers. I correctly get the numbers 4,13, but also get the numbers (4,55) (4,61) (3,64) (4,67) (4,73) (3,76) (4,79) (4,83) (4,89)

any comments? the way the puzzle was phrased i hoped to find a unique sollution. the program is in PERL (61 readable lines), and if anyone wants it leave your email and request in this thread.

Cheers

Gidon

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In response to this Extro... wrote:

quote:

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Here's the problem with (4,55)
If it were 4 & 55, Samantha knows the sum is 59. She can't discount the possibility that the numbers might be 6 & 53, in which case Paul would know from the product (6*53=318) that the numbers are 6 and 53. But she claims to know that Paul could not know the numbers.

Note: 318 = 6*53 = 3*106 = 2*159, but 106 and 159 are both over 100. If Paul knew the product to be 318, he would know the numbers are 6 & 53.

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Now that we have all been enlighted as to the fact that Paul said he had forgotten (and not that he couldn't have figured it out) 4,55 IS STILL A POSSIBLE ANSWER
You have all said so yourself...

so either way if P and S know the size of the town, or even if they don't there is STILL multiple answers that will fit the bill..

I haven't tried any of Gidon's others answers but I wouldn't doubt that all of his numbers work

So in conclusion:

There are at least two possible outcomes if they know the size of the town, and at least two possible outcomes if they don't know the size. (or can/can't figure out the answer in the first step).

There is a possible of 34 answers to this question.


Either way the question is flawed and it's not just that I interperated it wrong.

Oh and P.S. I'm changeing my answer back to 2,25 - because Like i said i live right next store in 2,24, so i knew this from the start.

Oh and before i forget, extro... i'm not saying your wrong (in fact just the opposite) you were just the only one who pointed out why 4,55 is not possible, which interestingly enough is the same reason that 4,13 is not possible (assuming that they know the size of the town).

So in the end nobody was right, the question was wrong .

I'm sure glad we FINALLY got to the bottom of this

[This message has been edited by HyToFry (edited 03-10-2000).]

[Ghost Post]

HyToFry: The problem with 4,55 still stands. Pauls first statement is that he forgot the two numbers. His second statement is that he can't figure them out (knowing their product and that they are in the range 2..100).

If the sum were 59, Samantha could not have known the numbers were not 6 and 53 before Pauls second statement, thus could not have known that Paul could not figure out the numbers (as he would be able to do if they were 6 and 53). So 4,55 is no good - Samantha's second statement wouldn't be true.

So far, I think we have three possible differences in interpretation of the puzzle:
1) "forgot" means "forgot", or it means "forgot, and can't figure out"
2) They either do or don't know the maximum house number is 100 (for first and second number)
3) When Sam says "I knew you couldn't figure out the numbers", she means she knew it from the beginning, or she means she knew it just prior to him saying it.
That's 8 possible variations in interpretation. I'll get back with answers for all 8, and it will be settled.

[HyToFry]

Extro... you found three, however there is really only one difference.

quote:

---

So far, I think we have three possible differences in interpretation of the puzzle:
1) "forgot" means "forgot", or it means "forgot, and can't figure out"
2) They either do or don't know the maximum house number is 100 (for first and second number)

---

1 and 2 have the same outcomes either way.
so they are basically the same, (either they did know the size, but couldn't figure it out) or they could have figured it out if they had known the size, if they couldn't figure it out, and didn't know the size it wouldn't matter.

[Ghost Post]

They both knew from the start that the town was a 100 by 100 grid.

Paul says "I forgot, but know the product, and that first is not greater than second".

Sam says "I forgot, but know the sum, and that neither is one"

Sam at this point (before his next statement) knows that Paul can't figure out the numbers. If the numbers were 6 and 53, he could figure it out now. If the sum were 59, Sam couldn't know what she does. So the sum is not 59, and the numbers can't be 4 and 55.

[HyToFry]

This is true, but also true for 2,26 (the other half of 4,13)

2+26 = 18 which has only two sets of prime numbers that can add up to it, 5,23 and 11,17

5*23 = 115 and so paul would know that it was 5,23 (as it couldn't possibly be 1,115) so sam can rule this number out.

11,17 works the same

since there is no other set of primes that can add up to 28 Sam would KNOW that Paul couldn't know the answer... (the reasoning is the same as it is for 4,55) it would be like saying that sam didn't know that paul might think the answer was 1,318. (the other set of 6,53)

See the light yet?

[This message has been edited by HyToFry (edited 03-10-2000).]

[HyToFry]

bradan said it best in one of his posts:

quote:

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(4,61), (16,73), (64,73) are not possibilities here because of the current size of the village!
If (4,61) was the address, => P=244. The only other possible address for P=244 is (2,122) which is not an address in the village. Therefore, Paul would get the address straight away. [unless of course he also forgot the size of the town!]

The same occurs for the possibility of:

(16,73):
P=1168 => other addresses are (4,292) & (2,584)... not in the village.

&(64,73):
P=4672 => other addresses are (16,292), (8,584), (4,1168), (2,2336)... also not in the village.

(4,13) remains the only address.

---

He proves my point with one sentance:

[Ghost Post]

You're kidding, right? Is that the light you're asking whether I see? That you're just jerking us around?

Please read my post above from 8:58 AM

It is no longer at all clear what point you are arguing.

Assuming "I forgot" means only that, and not also "I can't figure it out", and assuming they know the size of the town, then the only answer is 4,13.

It can't be 2,26, because Sam wouldn't be sure it isn't 5,23, in which case Paul WOULD know the numbers from the product (115). So 2,26 is NOT possible, because Sam knows the numbers are NOT 5,23. How could she know that if she only knows the sum (28)???

Do you agree?

[HyToFry]

No your wrong in the case that :

[Ghost Post]

"TOP OF PAGE"
YOU HAVE TO BE KIDDING!!!

Again, if it were 4 and 55, Sam would know the sum to be 59.

Paul says (his second statement), and I quote, "I can't figure out where the party is". He says this knowing the numbers are greater than 1 (fact) and no greater than 100 (same assumption as always, which leads to 4,13 as answer).

Samantha says "I knew that".

So, with the sum=59, she supposedly already knew (knowing only the sum, and that they both know both numbers are in the range 2..100, with the first <= the second) that Paul could not figure out the numbers.

Yes, I assume his first statement ("I forgot") means just that. But his second statement ("I can't figure out") means he can't figure it out.

If the sum were 59, how could Sam know that Paul couldn't "FIGURE OUT" the numbers, as he would be able to if the numbers were 6 and 53.

Again, after the first two statements (Paul and Sam saying "I forgot"), we (and THEY) know this:
A) 2 <= first <= second <= 100
B) Paul knows value of first*second
C) Sam knows value of first+second

NOW, AT THIS POINT, Sam knows that Paul "CAN'T FIGURE OUT" the numbers. She has not said so yet, but when she does (right after Pauls next statement), THIS is the point in time she is referring to! Paul will now say, and I QUOTE, "I can't figure out where the party is", to which she replies "I KNEW that". She knew it before he said it. She knows it NOW.

If the sum were 17 (4+13), this would be true. The only possible products (2*15, 3*14, 4*13 ... 8*9) ALL have multiple factorizations into first*second with in the range 2..100, with first<=second. She would know that Paul "CAN'T FIGURE OUT" the numbers (knowing fact A above, and any of the possible products 30, 42, 52, ... 72).

If the sum were 59 (4+55), this would be false, because 59 can be 6+53, and 6*53=318 HAS ONLY ONE FACTORIZATION given the constraints of fact A above, which fact Paul is well aware of (SAME ASSUMPTION as for 4,13). If the sum were 59, the product might be 318 (6*53), from which Paul could "FIGURE OUT" (given that he knows fact A above, which is THE SAME ASSUMPTION used for the 4,13 case) that the numbers are 6 and 53.

See the light? If not, goto "TOP OF PAGE".
If so, say "AHA!"

[This message has been edited by extro... (edited 03-10-2000).]

[HyToFry]

AHA!!!!!!!!!!!!!!!!!!!!!!! I GET IT .

However keep in mind that you are ASSUMING that Paul saying "forgot" means just that, and not "forgot and can't figure out", while at the same time you are ruling out the chance that Paul and Sam just happend to see the party. (which is about as far fetched).

Also you are Assumeing that Paul and Sam know the size of the town, "which it never says that it does", and third you are assuming that the puzzle knows that we can figure out the answer based on the information givin, however it never says this .

One other thing i noticed while takeing this puzzle TOTALLY literaly is this:
Paul never says in step one that he HAS forgotten the address, only that he is afraid that he forgot it, which isn't to say that he has, so Paul may have realized that he didn't forget it and Sam may have just rememebered.

So in conclusion i still think the puzzle should be more specific... and I'm pretty sure that you will agree.

Oh and one more thing, You agree that 2,25 is still a possible answer if forgot means forgot and can't figure out, and P and S know the size of the town right? (this will make me feel alot better just knowing that you agree with this one statement)



[This message has been edited by HyToFry (edited 03-10-2000).]

[HyToFry]

Okay extro... i have revisited your 3 possible scenerios and discovered that you were almost right, I can now see why "forgot" means just that and "not knowing the size of the town" is a different outcome, but using these two I can only see 3 possible interpertations of the puzzle
So far, I think we have three possible differences in interpretation of the puzzle:

quote:

---

1) "forgot" means "forgot", or it means "forgot, and can't figure out"
2) They either do or don't know the maximum house number is 100 (for first and second number)

---

I agree with these statements and this makes three possible outcomes:

1:They did not know the size of town, and "forgot" means just that (if forgot meant forgot and can't figure out it wouldn't matter because either way they couldn't figure it out) anywayz this makes 10 Possible outcomes, (4,13) (4,55) (4,61) (3,64) (4,67) (4,73) (3,76) (4,79) (4,83) and (4,89).

2:The DID know the size of town, and "forgot" means just that and not "forgot and can't figure it out", this makes ONE possible outcome, (4,13) And since we can ASSUME that the puzzle knows we can solve it, although it never says that we can (interestingly enough this is another possible outcome) this is probably going to be the right answer

3:They did know the size of the Town and forgot means forgot and can't figure it out. This makes 24 possible outcomes (2,25) (2,27) (2,42) (2,44) (2,54) (2,58)
(2,82) (3,25) (3,35) (3,45) (3,51) (3,69)
(3,74) (3,86) (4,74) (4,82) (5,35) (5,58)
(6,74) (7,58) (7,62) (10,58) (11,46) and (13,34).

The other criteria you have stated is impossible:

quote:

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3) When Sam says "I knew you couldn't figure out the numbers", she means she knew it from the beginning, or she means she knew it just prior to him saying it.
That's 8 possible variations in interpretation. I'll get back with answers for all 8, and it will be settled.

---

It is IMPOSSIBLE for S to know that P cannot figure out the answer before P makes his first statement. For this to be possible S would have to know that P knows the Product and has forgoten the Address. S doesn't have a clue about this until after P makes his first statement, so for all S knows P could figure it out based off the fact that P may or may not know the Address, and the fact that S doesn't even know that P knows the product until after finding out that P has forgotten the address.


4,13 is the only possible answer in these cases, ASSUMING that the puzzle knows we can solve it (which it doesn't say)

Thanks extro... for all of your help in getting to the bottom of this VIRIS of a puzzle... YOU ROCK!!!!!!!!!!!!!!!!!

oh ROCKING is a good thing (just thought i would throw that in, just in case English isn't your first langauge.



[This message has been edited by HyToFry (edited 03-10-2000).]

[Ghost Post]

Yes, 2,25 is one of 24 solutions under that interpretation.

I'm confident though that the "official" interpretation of 'forgot' will be that they forgot, with no implication about their ability to figure it out. That is a problematic interpretation were it real life though.

For instance, if Paul knew the two numbers were positive integers, and knew the product to be 1, could he possibly claim that he "forgot" the two numbers? If he knows the size of the town, can he claim to have forgotten the numbers if he knows the product to be 289 (17*17)?

Where is the dividing line between what one can reasonably say they forgot, even though they have enough information to figure it out with some thought, and what one can't resonably say they forgot given how easy it is to figure out from known information?

A brilliant mathematician may claim to have forgotten what the billionth or hundreth prime number is, but not the first, second or third prime number. How about the tenth, or twentieth?

[Ghost Post]

4 13 is the only solution.
don't ask me to write down the reasons!
(I began computation by hand trying the couples adding 11 or 17 or 23 or ... ; I suspected I made a mistake, because my numbers became to grow, so I had to program the problem on my PC (with Mathematica) and I discovered the 4-13 solution; it is not very polite to solve a game with a computer, but I had to do it!)

[Ghost Post]

The solution is 4,13.
(Don't ask me why, I am tired. I solved it last week, and I admit that I had to write down a program on my PC, because trying with paper and pen I made a mistake and jumped the right sum 17)


------------------

[Ghost Post]

The solution is 4,13.
(Don't ask me why, I am tired. I solved it last week, and I admit that I had to write down a program on my PC, because trying with paper and pen I made a mistake and jumped the right sum 17)