# Introductions

1. Rearrange the sentences below into a good conversation!

Alice: Hi Bob, how are you?

• Alice: I'm fine too, but my school kinda sucks.
• Bob: I'm fine, thanks.

Bob: Ouch, what happened?

Alice: Hi Bob, how are you?
Bob: I'm fine, thanks.
Alice: I'm fine too, but my school kinda sucks.
Bob: Ouch, what happened?

2. Cross out the wrong one between the two possible answers!

Alice: My cat was dead tonight.
Bob: (Congratulations! / I'm sorry to hear that.)
Bob: Don't be sad, let's go for lunch together. I'll treat you.
Alice: (Thanks / You're welcome), Bob.

Alice: My cat was dead tonight.
Bob: (Congratulations! / I'm sorry to hear that.)
Bob: Don't be sad, let's go for lunch together. I'll treat you.
Alice: (Thanks / You're welcome), Bob.

3. Fill in the blank with appropriate number!

Alice: Uhh.. What is $96$ divided by ___?
Bob: It's $24$.
Alice: Oh! I'm sorry, I mean divided by $6$.
Bob: Oh, it should be ___.

Alice: Uhh.. What is $96$ divided by $4$?
Bob: It's $24$.
Alice: Oh! I'm sorry, I mean divided by $6$.
Bob: Oh, it should be $16$.

# The Puzzle

Well, do all of them at once! :)

Alice: I'm thinking of a number, between $1$ and $10$ inclusive.

• Alice: Eh, $4$ possible answers? Hmm, whatever... Anyway, the answer for your question is (yes / no).
• Alice: No, it is not.
• Alice: (Yes / No).
• Bob: Is your number an odd number?
• Bob: Is your number a prime number?
• Bob: Is your number between $7$ to $10$ inclusive?
• Bob: Now I have $4$ possible answers. Let me ask you another question.

Bob: Hmm.. Now there are ___ possible answers. Which one is your number...
Alice: Huh, you are supposed to know my number after that last question. Well, indeed you can't deduce it with only your first two questions...
Alice: Oh shoot! I should have answered "no" for your (first / second) question!
Bob: Ah! OK then. Now I know that your number is ___.

Alice: I'm thinking of a number, between $1$ to $10$ inclusive.

Bob: Is your number a prime number?
Alice: (Yes / No).
Bob: Now I got $4$ possible answers. Let me ask you another question.
Bob: Is your number an odd number?
Alice: Eh, $4$ possible answers? Hmm, whatever... Anyway, the answer for your question is (yes / no).
Bob: Is your number between $7$ to $10$ inclusive?
Alice: No, it is not.

Bob: Hmm.. Now there are $2$ possible answers. Which one is your number...
Alice: Huh, you are supposed to know my number after that last question. Well, indeed you can't deduct it only with your first two questions...
Alice: Oh shoot! I should answer "no" for your (first / second) question!
Bob: Ah! OK then, now I know that your number is $1$.

Logic:

Bob asks three questions. Between 1 and 10, we know that...
there are 5 odd numbers: 1, 3, 5, 7, 9
there are 4 prime numbers: 2, 3, 5, 7
there are 4 numbers between 7 to 10, inclusive: 7,8,9,10

As Bob mentions having 4 possibilities at some point, the first question was either the prime one or the 7 and 10 one, and the answer was "yes". This means the answer to the last question was "no". Note that if the first two answers had been not odd and not prime (giving 4,6,8,10), then the answer to the first two questions would be "no", whereas Alice said at least one of them was "yes" (and should have been "no").

If we assume the first reply to be that the number is between 7 and 10, then it can't be odd since Bob would know if it is 7 or 9 from primality. So it has to be not odd and not prime.

Changing the answer to the first question still doesn't help to determine what the number is, but changing any other answer would requires changing a "no", not a "yes" as is said in the question! Therefore the first question has to be the primality one.

If we assume the first reply to be that the number is prime, then because Bob can't determine what it is, it must be odd and not between 7 and 10. (this allows for it to be 3 or 5).

Remembering Alice's mistake, if the mistake was on the second question, the number is prime, even and not between 7 and 10. This would allow Bob to deduce it using only 2 questions, so it must not be the case.

Finally, if the mistake was on the first question, the number is not prime, odd and not between 7 and 10: it must be 1.

• Was just about to answer the same thing. Well played ! Apr 20, 2018 at 9:08
• Great job! I'm going to point out a minor flaw: Bob having $4$ possible answers is not only because of it's a prime number, or it's a number between $7$-$10$. For example, another possible scenario is that the number is not odd and not prime: $4, 6, 8, 10$. Apr 20, 2018 at 12:00
• @athin correct me if I'm wrong, but if the number were neither odd nor prime, then Alice couldn't say "Oh shoot! I should answer "no" for your (first / second) question!", right? That sentence tells us she said "yes" to at least one of the first two questions. Or was your point that ffao didn't say this? Apr 20, 2018 at 15:50
• @LordFarquaad, yep you are correct. My point is that ffao didn't include this statement, ^^ Apr 20, 2018 at 15:59
• @athin I could have sworn I included that case... oh well, added now.
– ffao
Apr 20, 2018 at 19:48