Riddles/Brainteasers

[Pudge714]

GoT,

I really hate you.
I'm convinced you are intentionally screwing up my productivity.

[stealthcow]

pretty sure i got this. i really should be studying for my quantum test though.

answer below:
<font color="white"> im pretty sure the numbers are 3,8. mr P gets 24, so possible pairs are 2,12 3,8 6,4. he says he doesn't know the answer.

mr S gets 11 and of the possible products 2x9 = 18, 3x8 = 24, 4x7 =28 and 6x5 = 30, he knows each can be factored in more then 1 way. so he says i knew you wouldnt know the answer. and i dont know it either.

but if you look at the 3 pairs who multiply to 24, only one of them allows mr S. to say what he did. with S = 10, its possible that mr P got 25, so he would know the answer, and with S = 14 its possible P got 33 and would also know the answer.

so then Mr. P gets the answer, and Mr. S is able to deduce how Mr. P comes to the same answer</font>

[srjunkacct]

[ QUOTE ]
pretty sure i got this. i really should be studying for my quantum test though.

answer below:
<font color="white"> im pretty sure the numbers are 3,8. mr P gets 24, so possible pairs are 2,12 3,8 6,4. he says he doesn't know the answer.

mr S gets 11 and of the possible products 2x9 = 18, 3x8 = 24, 4x7 =28 and 6x5 = 30, he knows each can be factored in more then 1 way. so he says i knew you wouldnt know the answer. and i dont know it either.

but if you look at the 3 pairs who multiply to 24, only one of them allows mr S. to say what he did. with S = 10, its possible that mr P got 25, so he would know the answer, and with S = 14 its possible P got 33 and would also know the answer.

so then Mr. P gets the answer, and Mr. S is able to deduce how Mr. P comes to the same answer</font>

[/ QUOTE ]

Mr. S still cannot decide between (3,8) and (4,7) though. The product 28 allows Mr. P to make the same statements; 28 = 2*14 = 4*7, and he knows that 2+14 would not allow Mr. S to make the first statement, as 2+14=16=3+13 (two primes).

[Yeti]

stealth,

I just googled and it would appear that is wrong.

[EverettKings]

OK, final written solution to this problem:

[ QUOTE ]

Problem:

Three gods A , B , and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A , B , and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer in their own language, in which the words for yes and no are “da” and “ja”, in some order. You do not know which word means which.

[/ QUOTE ]

I'm not writing anything in white so just dont read any further if you don't want to know.

I. Translation

First, getting past this 'da'/'ja' garbage:[ QUOTE ]

The gods understand English, but will answer in their own language, in which the words for yes and no are “da” and “ja”, in some order. You do not know which word means which.


[/ QUOTE ]
Say you know you're talking to the Truth god, and you want to ask him "Do you like turkey?". He could say 'da' or 'ja' and you have no clue if he likes turkey or not. You want to find a way that he will say 'da' if he likes turkey and 'ja' if he doesn't like turkey. This is what you ask:

"Is at least one of the following true?
A) You like turkey AND 'da' means yes
B) You DONT like turkey AND 'ja' means yes"

<u>There are 4 cases:</u>

i. 'da' = yes and he likes turkey
In this case, criteria A is true so the god will want to answer yes, and will say 'da' (since that means yes).

ii. 'da' = yes and he doesn't like turkey
In this case, neither criterion are met so the god will want to answer no, and will say 'ja' (since 'da' means yes).

iii. 'ja' = yes and he likes turkey
In this case, neither criterion are met so the god will want to answer no, and will say 'da' (since 'ja' means yes).

iv. 'ja' = yes and he doesn't like turkey
In this case, criteria B is met and the god will want to answer yes, and will say 'ja' (since 'ja' means yes).

ZOMG! In both cases where he likes turkey(i/iii), he says 'da'. And in both cases where he doesn't like turkey(ii/iv), he says 'ja'. Thus by asking your question "Do you like turkey" with this modified logic, you have ensured that you get the answer 'da' if he wants to say 'yes', and you get the answer 'ja' if he wants to say 'no', regardless of the actual meaning of the words.

You must apply this technique to every question you ask a god. That way anytime you hear 'da' you know that he meant to say 'yes', and anytime you hear 'ja' you know that he meant to say 'no'.

Note also that the False god will always answer the opposite ('ja' if the question is true and 'da' if the question is false) and the Random god whatever, so this technique is still valid for them.

II. Interrogation

With that little problem out of the way there is still a full puzzle in itself to be solved -- who to ask what to figure things out. For simplicity I'm gonna ignore 'da' and 'ja' now and just use 'yes' and 'no', since all of our questions will be asked in a modified manner such that 'da' means yes and 'ja' means no.

We're looking at 3 gods that we will label (from left to right) A, B, and C. Denoting the gods' actual identities T, F, and R, we have the following possible orderings:

TFR
TRF
FTR
FRT
RTF
RFT

a. <u>The general idea</u>

Picking the first question is crucial. We'll ask god A since it doesnt matter which. Say, for example, we ask (in our roundabout manner discussed in part I) "Does 2+2=4?" (this isn't actually the question we want to ask, but just an example to demonstrate the problem). We know that the Truth god will say 'yes', the False god will say 'no', and the Random god will say whatever makes your life hardest (we have to consider worst case here). Either way, if you hear 'yes' then you know god A is NOT the False god (who would never give an honest answer), and if you hear 'no' then you know that god A is NOT the Truth god (who would never lie). Let's say we hear 'no', then we're down to four possibilities:

FTR
FRT
RTF
RFT

*IMPORTANT* After our next question (question 2), we MUST be down to 2 possibilities or fewer, since if we have 3 we can't possibly differentiate 3 states with one question (one binary number can represent up to 2 values).

Unfortunately, this particular question won't work, since there is no question you can ask to any person to guarantee that you'll be down to two options. If you ask god B a question, for example, then the random god might answer the same way as the Truth god would, and you can only eliminate the last possibility (still leaving you with three). I know I didn't explain that well, but try asking questions to A B or C and you will see that the random guy can always answer in such a way that, in the worst case, we still have 3+ possiblilities remaining. So now what?

b. <u>The right question</u>

You have to ask a question to eliminate the Random god from one of the slots. There's probably plenty of questions that will do this, but the one that I came up with is:

1) (To God A) "Is God B more truthful than God C?"

What I mean by 'more truthful' is the following: Truth is more truthful than Random, who is more truthful than False. So if gods B and C are (respectively) TR, TF, or RF then the honest answer is 'yes', and if gods B and C are FT, RT, or FR then the honest answer is 'no'. Let's look at our possible combinations and how god A will answer:

1. TFR -- NO -- Truth will be honest. F is NOT more truthful than R.
2. TRF -- YES -- same idea
3. FTR -- NO -- False will lie. T is definitely more truthful than R, so F is dishonest and says no.
4. FRT -- YES -- same idea
5. RTF ??? Could be yes or no
6. RFT ??? Could be yes or no

If the answer that we hear is 'no', then we can rule out 2 and 4 for sure and are left with:

1. TFR
3. FTR
5. RTF
6. RFT

If the answer that we hear is 'yes', then we can eliminate possibilities 1 and 3. This leaves us with:

2. TRF
4. FRT
5. RTF
6. RFT

I'll only solve the 'yes' case since you can figure out the 'no' case in the same way.
In the remaining 4 possibilities above, notice that god C now cannot be random! This is a huge help, as we can now determine god C's true identity by asking him our second question:

2)(To god C)"Does 2+2 equal 4?"

We know the answer is always true, so if the god says 'yes' then he is the Truth god, and if he says 'no' then he is the False god. Let's say he says 'yes' (you can figure out the other case easily), this means he is Truth and we are down to:

4. FRT
6. RFT

So since we know Truth will always be straight with us, we can ask him our final question:

3)(To god C)"Is God A the random god?"

If he says yes we have case 6, if no we have case 4. Yadda yadda you can extrapolate and figure out the cases that I skipped.

Let me know if I didn't explain something clearly or you want to see the 3 questions spelled out in their complicated form.

This was a sweet puzzle.



Everett