### Brain Teasers

# The Total is 14, Part III

Mr. Simkin, the new math teacher at school, was impressed by his students' ability to solve logic puzzles. He pulled aside three more students, and handed them each a sealed envelope with a number written inside. He told them that they each have a positive integer, and the sum of their numbers was 14.

Manny, Moe, and Jack each opened their envelopes. Mr. Simkin asks Manny if he knows anything about the numbers the other two are holding, and Manny says, "I know that Moe and Jack are holding different numbers."

Moe answers, "IN THAT CASE, I know that all three of our numbers are different."

Jack thinks for a bit, and then says, "Now I know all of our numbers."

Mr. Simkin turns to the class and asks if anyone in the class knows the numbers. Gretchen's hand shoots up into the air, and after waiting for a while to see if anyone else will get the answer, Mr. Simkin calls on Gretchen.

What numbers does she say they each are holding?

Manny, Moe, and Jack each opened their envelopes. Mr. Simkin asks Manny if he knows anything about the numbers the other two are holding, and Manny says, "I know that Moe and Jack are holding different numbers."

Moe answers, "IN THAT CASE, I know that all three of our numbers are different."

Jack thinks for a bit, and then says, "Now I know all of our numbers."

Mr. Simkin turns to the class and asks if anyone in the class knows the numbers. Gretchen's hand shoots up into the air, and after waiting for a while to see if anyone else will get the answer, Mr. Simkin calls on Gretchen.

What numbers does she say they each are holding?

### Answer

Manny has a 3, Moe has a 2, and Jack has a 9.From Manny's statement, we can deduce that his number is odd. Since Moe did not know that they were all different until Manny said that, we know that Moe is not holding a 7, 9, or 11. (Otherwise, he would have already known they were all different.) If he were holding a 1, 3, or 5, he would not be able to be sure his number was different than Manny's. If he were holding a 4, 8, or 12, he could not know that Manny and Jack didn't have the same number, since there are odd pairs that would bring the total to 14. Therefore, Moe must be holding a 2, 6, or 10.

There are 12 triples for (Manny, Moe, Jack) that satisfy all three statements. They are:

(1, 2, 11)

(1, 6, 7)

(1, 10, 3)

(3, 2, 9)

(3, 6, 5)

(3, 10, 1)

(5, 2, 7)

(5, 6, 3)

(7, 2, 5)

(7, 6, 1)

(9, 2, 3)

(11, 2, 1)

Since Jack can reason flawlessly, he knows these are the possibilities. In order to make his statement, his number has to be a unique solution. Therefore, he must be holding a 9 or an 11. If he were holding 11, though, he would have known from Manny's statement that Manny had a 1, Moe had a 2, and he had an 11. Since he didn't know them all until after Moe spoke, he must have a 9, leaving Manny a 3 and Moe a 2.

Mr. Simkin suggests to Gretchen that she may have a career in law enforcement if she can further hone these impressive deductive reasoning skills.

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