### Brain Teasers

# The Total is 14, Part I

Mr. Simkin, the new math teacher at school, pulled aside three students, and handed them each a sealed envelope with a number written inside. He told them that they each had a positive integer, and the sum of their numbers was 14.

Sage and Rosemary opened their envelopes, and on seeing his number, Sage said, "I know that Rosemary and Tim have different numbers."

Rosemary replied, "I already knew our numbers were all different, and now I know everyone's number."

Tim said, "Then I do, as well," even though he never opened his envelope.

How could he have known? What numbers did everyone have?

Sage and Rosemary opened their envelopes, and on seeing his number, Sage said, "I know that Rosemary and Tim have different numbers."

Rosemary replied, "I already knew our numbers were all different, and now I know everyone's number."

Tim said, "Then I do, as well," even though he never opened his envelope.

How could he have known? What numbers did everyone have?

### Answer

Sage had a 1, Rosemary had an 11, and Tim had a 2.From Sage's statement, he had to have an odd number. He knew that the sum of all three numbers was even, so if his number was odd, one of the other two numbers would have to be odd, and the other even, leaving them unequal.

Rosemary could deduce this, too, so when she saw an 11, she knew that Sage had a 1 and Tim had a 2. If she had seen a 12, she would have known that the other two boys each had a 1, which is contradicted by her saying that she already knew they all had different numbers. If she saw any number other than an 11 (say 9, for example), she would not have know which odd number Sage held. (In our example, he could have had a 1 and Tim a 4, or he could have had a 3 and Tim a 2).

Tim, understanding all of this, therefore knew their numbers.

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