### Brain Teasers

# The Total is 14, Part II

Mr. Simkin, the new math teacher at school, was impressed by their 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.

They each opened their envelope and inspected their numbers. Snap said, "I know that Crackle and Pop each have a different number."

Crackle replies, "I already knew that all three of our numbers were different."

Pop says, "Now I know what all three of our numbers are."

Mr. Simkin turns to the class to ask if anyone else knows the numbers. Only Gretchen raises her hand, and she correctly identifies what each person is holding. How did she know?

They each opened their envelope and inspected their numbers. Snap said, "I know that Crackle and Pop each have a different number."

Crackle replies, "I already knew that all three of our numbers were different."

Pop says, "Now I know what all three of our numbers are."

Mr. Simkin turns to the class to ask if anyone else knows the numbers. Only Gretchen raises her hand, and she correctly identifies what each person is holding. How did she know?

### Answer

Snap's statement implies that his number is odd. Crackle's statement implies that not only is his number odd, but it is also greater than or equal to 7. There are only six combinations that satisfy these conditions. Namely, for (Snap, Crackle, Pop) they are:(1, 7, 6)

(1, 9, 4)

(1, 11, 2)

(3, 7, 4)

(3, 9, 2)

(5, 7, 2)

Since Pop can reason perfectly, he knows these are the possible combinations. In order for him to know the other two, he must be holding a 6, since he knew the solution had to be unique. Therefore, Snap has a 1, Crackle has a 7, and Pop has a 6.

And, as usual, Gretchen solidifies her place as a top student.

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