Two numbers, two operations, two compititors! one liar!

Logic Level pending

Two logicians are in a competition to correctly pick two distinct integers between 2 and 7 inclusive. One of them, Sam, is given the sum of the two numbers, while the other, Dan, is given the difference of the two numbers. Neither one knows what the information given to the other is. Whoever gets the two numbers correct would receive a well deserving prize.

Now, the following conversation takes place:

Sam: I don't know the numbers.
Dan: Neither do I.
Sam: I wasn't sure whether you knew the numbers from the very beginning before our conversation took place.
Dan: Oh, I was sure that you didn't know the numbers before our conversation took place.

At the end of this conversation, they each conclude what those two numbers are, and submit their respective answers to the judges. Then the results are released, and Sam didn't get the correct pair of numbers, while Dan did.

Sam is furious because he is sure that he didn't make any errors and complains to the judges. Everything is settled when Dan admits that his very last statement was a deliberate lie so that Sam would not get the right answer, as it is prohibited to lie during the conversation.

Can you determine the two integers that were written by Dan?

Let \(A\) and \(B\) be these two integers such that \(A< B\), then submit your answer as \(A + \dfrac{B}{10} \).


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