Problem:
Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that
the numbers are all different,
they sum to , and
they are in increasing order, left to right.
First, Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then Tracy looks at the number on the rightmost card and says, "I don't have enough information to determine the other two numbers." Finally, Stacy looks at the number on the middle card and says, "I don't have enough information to determine the other two numbers." Assume that each person knows that the other two reason perfectly and hears their comments. What number is on the middle card?
Answer Choices:
A.
B.
C.
D.
E. There is not enough information to determine the number.
Solution:
There are eight ordered triples of numbers satisfying the conditions: , , , , , , , and . Because Casey's card gives Casey insufficient information, Casey must have seen a or a . Next, Tracy must not have seen a , , or , since each of these would enable Tracy to determine the other two cards. Finally, if Stacy had seen a or a on the middle card, Stacy would have been able to determine the other two cards. The only number left is , which leaves open the two possible triples and .