An old Bob Kraus puzzle from MathPuzzle

[thatsmyusername]

http://www.mathpuzzle.com/krauspuz.html

[MNOWAX]

I know the answer, so I will allow some of the other GL'ers have a chance.
I have no f'in clue in other words.
_________________
The Man The Myth The Legend
MNOWAX

[thatsmyusername]

Can you maybe give us a hint what it means for the students to reject the hint offers? I tried the "it doesn't help because we already know the answer to that hint" route, for example, but that does not result in a solution.

I also tried the "it actually helps a little bit, but I'd still need more information before I can solve it, so I guess in a short-sighted way that doesn't really help" interpretation, and that got nowhere too.

Also, I'm not sure what the policy is regarding cross-linking to other forums, but if you google search the subject of the thread, you'll see my post discussing this problem elsewhere.

[MNOWAX]

highlight my entire post, and you'll get a hint on the answer.
_________________
The Man The Myth The Legend
MNOWAX

[ralphmerridew]

"would it help?" means "would you be able to answer all three questions?"

[Zag]

I agree that the question is poorly worded. It isn't even clear to me that the students even are told which envelope they recieved -- they might just have a number! Of course, then I think it is clearly impossible.

[3iff]

For the original page, it's posed as a logic problem...and I'm guessing it must be solvable.

N must be from 111 to 999 with the limitation that subsequent digits being are at least as high as the previous digit.

As they are allowed to use computers, they should be able to calculate which values of N are allowed, and then compute all possible answers for V=F(N). This should allow certain combinations to be eliminated as each person can eliminate possibilities (from their list).

From the "No" answers to the offered hints, each must realise that the others cannot benefit from the hints.

Eventually...there must be a combination of values that are unique. Yes??

[ralphmerridew]

If you add in typical assumptions, it become solvable (though some questions turn out to give no information to anyone), and there's a logically invalid method that unfortunately gets the correct answer.

[Zag]

Really, rm?

I've come down to it being one of these combinations

[ralphmerridew]

Check your work. That is not the correct set of possibilities.

[Zag]

[ralphmerridew]

I took "would it help?" to mean "would you be able to immediately answer all three questions?". (If I took it to mean "would you be able to prune your list of possibilities?", then somebody would have answered "yes" at 1:31.)

[Zag]

Gaah. I suck at ignoring stuff.

So, rather than Excel, I wrote a Java program to do the analysis. I realized what my mistake was before, but I still end up with both of the choices

[ralphmerridew]

Your program was too aggressive in eliminating cases; unless my program was wrong, you eliminated way too many cases.

[Zag]

LOL. Probably.

I wasn't sure with your comment here:

[ralphmerridew]

I'd have to check my program's source, but I'm pretty sure it was "two cases, one with answer yes and one with answer no".

[thatsmyusername]

[ralphmerridew]

Well, I found a solution assuming that students do know which envelope they have.

[thatsmyusername]

So at 1:25 you only have 625 possibilities left? You eliminated [N=899, F=SSQ, V=226] even though a person that sees 226 does not know if that number is N or V?

I already tried that route and I couldn't find a solution, so I'd be curious where you and I differed. I'm still exploring this N/V-ambiguous route. It does look promising.

[thatsmyusername]

I didn't get anywhere with N/V being ambiguous, however someone at another forum brought up the possibility that F is also given as a number 1..5.

Thus, someone who sees the number "1" in his paper can't tell if that's V or F.

I'd be curious how you claimed to have found a solution without these seemingly-forced complexities where all others have failed. Would you care to just explain and spoil it for us?

[Zag]

I got pretty close, and rm is a lot smarter than I am. You should be less suspicious.

Anyway, I don't see how making things more ambiguous would make it more solvable.

[thatsmyusername]

Well, I was hoping that making a number ambiguous between N and V would preserve triples that may later turn out to be unique at 2:00 (and therefore is the solution).

[thatsmyusername]

OK I'm going to detail how far I'd get if:
- I assume that envelopes are clearly labeled (so to speak)
- A hint is only "helpful" if it allows them to immediately single-out the solution

After 1:00 - 825 possibilities
After 1:20 - 625 left, 200 removed because V would've solved it
After 1:30 - 625 left ("wasted" round)
After 1:31 - 585 left, 40 removed because V + hint would've solved it
After 1:40 - 584 left, [899, LCM, 72] would've been solvable by N
After 1:41 - 572 left, 12 removed because N + hint would've solved it
After 1:50 - 572 left ("wasted" round)
After 1:51 - 113 left (374/180/204 would've been solvable by N/F/V + hint)

Among these 113, no unique value in N, F, or V.

Comments?

[ralphmerridew]

Well, I checked my source and found that my program actually used the rule of "Teacher told them 'None of you would be able to solve it given that information.'".

I found the author's orignal site, and my answer (derived with completely wrong logic) turned out to be correct. Unfortunately, he only gives the raw answer with no explanation whatsoever, so I don't know what the right explanation is.

[Zag]

[ralphmerridew]

My program essentially treated

Teacher: If I told you ____, would it help?
Students: No.

as if it were:

Teacher: Even if I told you _____, none of you would be able to answer the questions.

So for the case

[Zag]

Thanks! That clears it up. I had started with that approach, when I was just doing it manually in Excel, but then realized about three deletes into it that it didn't make sense as the question was worded. I'll bet, though, that it is how the original puzzle author also did it, if you ended up with the same answer he did. There's no reason to think that these puzzle authors are infallible.

[lostdummy]

Just noticed there are some new problems on forum, and tried this one - but my first fast attempt did not exactly solve it ;p

I remain with 2 possible solutions for each of them:

1:00 - Students begin working. Remains 825
1:20 - Teacher asks if anyone had found the answers. No one had. Remains 625
1:31 - Teacher asks if it would help if he told them if N was odd or even. All 3 say no. Remains 459
1:41 - Teacher asks if it would help if he told them if V was odd or even. All 3 say no. Remains 380
1:51 - Teacher asks if it would help if he told them the sum of N and V. All 3 say no. Remains 4

Which leaves 4 possible cases:
[242]: N=279 , F=PROD(N) , V=126
[254]: N=345 , F=PROD(N) , V=60
[737]: N=279 , F=LCM(N) , V=126
[749]: N=345 , F=LCM(N) , V=60

Actually, those who get N or V would be able to guess other persons N or V, but since PROD and LCM both result in same numbers here, they wouldn't be able to guess function.

Therefore, since it appears from above posts that there is unique solution here, I wonder which restriction I missed ;p

[thatsmyusername]

You pruned too aggressively. The answer is supposed to be SSQ(336)=54, which is one of the 113 possibilities I have left.

How do you go from 625 at 1:20 to 459 at 1:31? Maybe you pruned the 3 columns with sequential dependency instead of in parallel?

[lostdummy]

Nope, if anything I pruned too conservatively. And what would be "sequential dependency" anyway in this case?

Take example (with incorrect numbers, just for example) of all remaining combinations for value=6:

1) N=123, F=* , V=6
2) N=123, F=*, V=6
3) N=224, F=*, V=6

After "N+V" question, I presumed that at least 3rd row should be pruned, since if someone holds value of "6", if N+V happened to be 230 it would uniquely define result.

With that I get 4 results remaining after N+V step - those that i listed above.

BUT this SHOULD be even more aggressively pruned, since in above example anyone with value "6" would answer YES to "N+V" question (if answer/hint happen to be N+V=230 it solves problem, so it "would help").

With such pruning (and i consider it more logical than above example) I remain with NO SOLUTION possible, that is to say, after N+V question number of remaining possibilities is 0(Zero) .

I noticed that RM mentioned considering "would help" as something along "even if i tell you , it wont help you". While that is much different from stated problem, since then teacher use information that only he knows, it is close to above example when I remain with 4 solutions. There, I prune only 3rd row, comparable to considering that teacher would know it would help them to get N+V if someone had 3rd row.

[lostdummy]

I find that even if I skip all other questions, and only use that "N+V" question, I have same two solutions:

a) if i interpret question as "even if i told you N+V, it wouldnt help you", I get results that I posted above
b) if i interpret it as "students who MIGHT benefit from N+V (if it happen to be certain value) would answer YES", then problem does not have solution - 0 remains possible.

BTW, is there any proof for SSQ(336)=54 ?

Doing quick analysis only from N+V question, it goes like this:

1) at start we have (among those 825 combinations):
2x (F=SSQ=3) and (N+V=390) :
N=269 , F=3 , V=121
N=336 , F=3 , V=54

1x (N=269) and (N+V=390):
N=269 , F=3 , V=121

2) if someone had N=269 and teacher told that "N+V=390", he would immediately know solution (other two numbers are 3 and 121). So combination "N=269 , F=3 , V=121" is removed from possible solutions

3) now we have only one combination with (F=SSQ=3) and (N+V=390) :
N=336 , F=3 , V=54

Since teacher knew that if he told 390, after step2 it would pinpoint only remaining possible solution, he could not say that "N+V would not help" - therefore we must remove also "N=336 , F=3 , V=54" as possible solution.



That means solution SSQ(336)=54 get removed after only 2 steps if "N+V" question is considered as "Teacher: it would not help you even if i tell you N+V". And so far that interpretation of "would help" is best that I can see, since it leaves us with 4 possible solutions, while more logical interpretation (b) leaves us with no possible solutions at all.

[lostdummy]

ok, I think i have one interpretation where I can get single solution.

If "would it help" is considered to be "even if i tell you N+V, you still would not be able to solve problem without additional question/info" then it can be solved. That "without additional question/info" means they could not be certain what is solution unless next "do you have answer" question is asked.

That would prevent "recursive" removing of possible solutions, and will result in 14 possible solutions after "N+V" question:

N=118 , F=2 , V=8
N=235 , F=2 , V=30
N=279 , F=2 , V=126
N=345 , F=2 , V=60
N=336 , F=3 , V=54
N=117 , F=5 , V=7
N=119 , F=5 , V=9
N=137 , F=5 , V=21
N=157 , F=5 , V=35
N=179 , F=5 , V=63
N=235 , F=5 , V=30
N=279 , F=5 , V=126
N=345 , F=5 , V=60
N=358 , F=5 , V=120

Out of those, only one combination will enable all 3 of them to say "I know answer" on last question, and that is "N=336 , F=3 , V=54"