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 $A + \dfrac{B}{10}$ .

The answer is 4.5.

1 solution

Abdou Abdou
Aug 28, 2016

Solution:

This is a dataset gouped by common sums/differences.

Differences	=05	=03	=04	=01	=02
				2 , 3		Sum=05
					2 , 4	Sum=06
		2 , 5		3 , 4		Sum=07
			2 , 6		3 , 5	Sum=08
	2 , 7	3 , 6		4 , 5		Sum=09
			3 , 7		4 , 6	Sum=10
		4 , 7		5 , 6		Sum=11
					5 , 7	Sum=12
				6 , 7		Sum=13

This table shows classified informations arrowed by similar sums, vertically divided by differences, As the summy preceded by the declaration that he doesnt know, everyone got in mind that :

The sum isnt {5 or 13}.

The other followed he dosnt know either, now we can tell:

The column difference = {5} is excluded

The summy was not sure about that diffy knows/or doesnt, he doubted he would know immediately, he suspected he had difference=5, so:

The sum is 9

At that point, diffy knows the number, despite he ensured the correct answer, he wanted to set his opponent up, so he diverted him toward The difference =3, because :

The difference = 1 is the only difference set in the set of sum=9 that contains other couples (2,3) and (6,7) where summy could have identified them immediately once told, saying he was sure that summy didnt know, excludes this choice which diffy lied about.

@Calvin Lin