Three-digit numbers

Logic Level 2

Alice and Bob were separately told consecutive three-digit numbers. Then they had the following conversation:

Alice says, "I don't know Bob's number."
Bob says, "I still don't know Alice's number."
Alice says, "I still don't know Bob's number."
Bob says, "Now I know Alice's number!"

If they told the truth, what is the sum of Alice and Bob's numbers?

207 1991 208 We don't have enough information to say. 500

2 solutions

Maggie Miller
Aug 8, 2015

Alice's first statement tells us that Alice's number is not 100 or 999.

Bob's first statement tells us that Bob's number is not 100,101,998,999.

Alice's second statement tells us that Alice's number is not 100,101,102,997,998,999.

Bob's second statement tells us that one of the following is true:

Alice's number is 995 and Bob's is 996.
Alice's number is 104 and Bob's is 103.
Alice's number is 996 and Bob's is 997.
Alice's number is 103 and Bob's is 102.

Thus, $\boxed{\text{we don't have enough information to say}}$ what the sum of the two numbers is.

Hi Maggie, I think I might have found a way to prove that the answer is 207. Tell me what you think.

* The same situation could happen if Alice and Bob had respectively 997 and 996. Because 1993 is not in the choices, we know this scenario is not conceivable. *

Let’s imagine Situation 1 in which Alice is 102 and Bob is 101. Turn 1 : Alice, thinking that Bob is either 101 or 103, doesn’t know. Turn 2 : Bob, being 101, knows that Alice is either 100 or 102. He also knows that that if Alice had had 100, she whould have known the answer on the first turn. Hence, Bob now knows that Alice’s number is 102.

Let’s imagine Situation 2 in which Alice is 102 and Bob is 103. Turn 1 : Alice, thinking that Bob is either 101 or 103, doesn’t know. Turn 2 : Bob, being 103, knows that Alice is either 102 or 104, but he doesn’t know which. Turn 3 : Alice knows that if Bob had had 101, Situation 1 would have happened and he would have known the answer on Turn 2. Hence, Alice now knows that Bob’s number is 103.

Let’s imagine Situation 3 in which Alice is 104 and Bob is 103. Turn 1 : Alice, thinking that Bob is either 103 or 105, doesn’t know. Turn 2 : Bob, being 103, knows that Alice is either 102 or 104, but he doesn’t know which. Turn 3 : Alice still doesn’t know. Turn 4 : Bob knows that if Alice had had 102, Situation 3 would have happened and she would have known the answer on Turn 3. Hence, Bob knows that Alice’s number is not 102, but 104.

Situation 3 is the situation in the problem. We know that the two numbers are 103 and 104, and that the answer is 207.

To elaborate on the problem, if the two numbers are (99+N, 100+N) or (1000-N, 999-N), the earliest the answer can be known by one of the characters is on Turn N, after (N-1) turns of them not knowing. If N is even, the person with the largest number must answer on Turn 1, otherwise they’re gonna need one more turn. If N is odd, the person with the smallest number must answer on Turn 1, otherwise they’re gonna need one more turn. Usually, even if we know the number N of turns, we wouldn’t be able to know if the two numbers are (99+N, 100+N) or (1000-N, 999-N). The 4 choices available in this problem solve this for us, offering us only one of the two possibilities.

Louis LeLouis - 5 years, 8 months ago

I think using missing answer choices to rule out answers isn't quite valid - for example, if I asked you "What is the solution set to $×^2=1$ over the real numbers" and gave you the options " $\{1\}, \{2,-2\}$ ," then neither would be right - the first wouldn't be correct because it was somehow less wrong. But I like the thought! :)

Maggie Miller - 5 years, 8 months ago

I've always seen the choices as part of the information given in the problem, and I'm pretty sure I've seen some problems built with this in mind on Brilliant, this should be clarified for the whole website. Thanks for your answer nonetheless !

Louis LeLouis - 5 years, 8 months ago

Why can't Alice be 103 and Bob 102? Alice is 103, he says I don't know If Bob was 102, he also would say I don't know thinking if Alice had 101, then Alice would know that Bob has 102, but if Alice still didn't know, then he had to have 103. So after Alice says for the second time he doesn't know, then Bob says he knows that Alice has 103.I hope I'm making sense.

vishwas u - 5 years, 10 months ago

You're right - edited for correctness.

Maggie Miller - 5 years, 10 months ago

Why is this problem so underrated?

Hrishikesh Kulkarni - 5 years, 9 months ago

Lygia Filgueiras
Aug 17, 2015

to add, I need numbers , in this conversation doesn't have number, so we don't have enough information to say what is the sum of Alice and Bo's numbers

Well, this is a play on a popular type of problem where you can determine the numbers even without any given. Try the famous Cheryl's Birthday problem .

Maggie Miller - 5 years, 10 months ago

Three-digit numbers

2 solutions

0 pending reports