In a Ted-Ed video, Brilliant is promoted with a problem about Ron and Harry. To know the solution, you have to come here to brilliant.org. The problem is the following:

"Harry and Ron told me separately how many times they watched the movie Titanic. I told them "you guys both watched it, but one of you watched it once more than the other". Then they had the following conversation:

Harry: Did you watch Titanic more than I did? Ron: I have no idea. Harry: Me neither. Do you know now? Ron: Yes, indeed! Harry: Really? Then so do I!

What is the sum of the possible number of times Ron watched Titanic?" End of the problem.

The solution says that the sum is 5, which means that one of them has watched it 2 times and the other, 3. But I do not agree with the solution, and this is the reason:

Let's assume that Harry has watched it twice and Ron 3 times. If that's the case, Harry knows Ron could have watched it either 1 time or 3 times. After the first question, Ron replies that he doesn't know, which means that the option where Ron have watched the film once is not true (as he would have replied that he didn't watched it more than Harry), which leads Harry to know that the only possible option is that Ron had watched it 3 times. And if that is the case, then he would have said: —I do know! — Instead of replying: —Me neither.

If we go with the other hypothesis; Harry has watched it 3 times and Ron twice. Ron knows that Harry has watched the film either 1 time or three times, so when Harry asked if Ron have watched it more than him, he can assume that Harry haven't watched the film just once as Harry wouldn't have had the need of asking because he would have known the answer. That is why Ron can assume that Harry has watched it 3 times; being ron's answer: I do know. Instead of: I have no idea.

If we follow this line of thinking (similar to the famous black and white hats problem) we can realize that the correct answer is 7, having them watched Titanic 3 and 4 times. Not 2 and 3.

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## Comments

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TopNewestInitially both know the other one watched it aleast once. If one of them has watched it once then it is sure for him that the other one watched it twice. But Ron finds Harry cannot find the answer and realises Harry hasn't watched just once. By this Ron was able take away one of the possibilties and finds the answer. This would be possible only if Ron watched it twice, Harry now know Ron has watched twice.Ron must have taken away 1and knew that harry has watched it thrice.

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I am struggling because there is clearly a logical hole in this. You cannot just assume Ron seen it 3 times, and Ron cannot just assume Harry seen it 2 times. Here is why.

An example is to turn to your best friend and ask him/her if he/she has watched Harry Potter more than you. Without stating numbers, find out who watched it more.

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If they had watched it that many times then there would have been some additional conversation because they would have needed some additional information to get to the conclusion.

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No, the answer is correct. I think you misunderstood the question. It is not the sum of times Harry and Ron watched the film, but the sum of the possible number of times Ron did. The initial answer is 0. Now let's assume X is a positive integer. If all the statements in their conversation hold, add X to the answer. So the final answer is the sum of all X-s that match the flow of their conversation. The only ones are 2 (Harry watched 3 times) and 3 (Harry watched 4 times).

Also your statement "...so when Harry asked if Ron have watched it more than him, he can assume that Harry haven't watched the film just once as Harry wouldn't have had the need of asking because he would have known the answer." is incorrect, because it is clearly stated in the problem that "Harry's initial question does not convey any information about what he knows.", so the case with Ron = 2 and Harry = 3 is correct.

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