Brilliant Students
Calvin, Zandra, and Eli are students in Mr. Silverman's math class. Mr. Silverman hands each of them a sealed envelope with a number written inside.
He tells them that they each have a positive integer and the sum of the three numbers is 14. They each open their envelope and inspect their own number without seeing the other numbers.
Calvin says,"I know that Zandra and Eli each have a different number."
Zandra replies, "I already knew that all three of our numbers were different."
After a brief pause Eli finally says, "Ah, now I know what number everyone has!"
What number did each student get?
Format your answer by writing Calvin's number first, then Zandra's number, and finally Eli's number. For example, if Calvin has 8, Zandra has 12, and Eli has 8, the answer would be 8128.
The answer is 176.
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1 solution
First let's analyze Calvin's statement: "I know that Zandra and Eli each have a different number" From this statement we can conclude that Calvin has an odd number, only this will guarantee that Zandra and Eli have different numbers. (Only an odd number and an even number added to an odd number gets an even number in the end.)
Now, let's analyze Zandra's statement: "I already knew that all three of our numbers were different." This means she has an odd number and that the number must be high enough so that Calvin can't have the same odd number. Therefore she has to have either 7, 9, or 11.
Finally, let's analyze Eli's statement: "Now I know what all three of our numbers are." We have to look at all the possibilities left and then see in which case Eli can be absolutely certain what number the others have. The possibilities still left are:
Zandra has 11, Calvin must have 1, and Eli must have 2.
Zandra has 9, Calvin has either 1 or 3, in which case Eli must have 4, or 2 respectively.
Zandra has 7, Calvin has either 1, 3, or 5, in which case Eli must have 6, 4, or 2 respectively.
Only when Eli has 6, can he be certain about the number the others got, so he must have 6. When Eli has 6, Zandra must have 7, and Calvin must have 1. The answer is therefore 176.