Expert: Paul Klarreich - 12/2/2006

Question
Amy, Carl and Amanda are three students.  Only one of them is an AP Scholar.  Each of them in turn makes the following truthful statements.

Amy says:        1. If I am not the AP Scholar, I didn¡¦t get a 5 on the AP Calculus exam.
       2. If I am the AP Scholar, I did get a 5 on the AP Calculus exam.  

Carl says:         1. If I am not the AP Scholar, I didn¡¦t get a 5 on the AP Physics exam.
       2. If I am the AP Scholar, I did get a 5 on the AP Calculus exam.  


Amanda says:   1. If I am not the AP Scholar, I didn¡¦t get a 5 on the AP Calculus exam.
       2. If I am the AP Scholar, I did get a 5 on the AP Physics exam.  

If in addition to the above statements, it is known that the AP Scholar is the only one who got a 5 on one of the exams and the only one who did not get a 5 on the other exam, which on is the AP Scholar?  Explain clearly in complete English sentences why you are certain that your answer is correct.


Answer
Questioner:  Ariel
Category:  Advanced Math

Subject:  Reasoning puzzles.
Question:  Amy, Carla, and Bella are three students.  Only one of them is an AP Scholar.  

Each of them in turn makes the following truthful statements.

Amy says: 1. If I am not the AP Scholar, I didn't get a 5 on the AP Calculus exam.
2. If I am the AP Scholar, I did get a 5 on the AP Calculus exam.  

Bella(not Amy) says: 1. If I am not the AP Scholar, I didn't get a 5 on the AP Calculus exam.
2. If I am the AP Scholar, I did get a 5 on the AP Physics exam.  

Carla says:   1. If I am not the AP Scholar, I didn't get a 5 on the AP Physics exam.
2. If I am the AP Scholar, I did get a 5 on the AP Calculus exam.  

If in addition to the above statements, it is known that the AP Scholar is the only one who got a 5 on one of the exams and the only one who did not get a 5 on the other exam,  which is the AP Scholar?  
...............................................
Hi, Ariel,

I don't have a brilliant insight for you, but you can usually handle this type of problem by making assumptions and seeing if they lead to a contradiction or inconsistency.

(I changed some names, so we have A,B,C as the students, and all female.)

You state that on one of the exams, there was exactly one '5' and on the other there were exactly two '5's

Suppose A is the scholar, then from her second statement:

A got 5 on the Calc, therefore did not get 5 on the Phys, and B,C did not get 5 on Calc, and B,C did get 5 on Phys.

B did not get 5 on C.  -- OK.
C did not get 5 on P   -- INCONSISTENT.

A is not the scholar.

...............................

Suppose B is the scholar, then from her second statement:

B did get 5 on the Phys, therefore did not get 5 on calc, and A,C did not get 5 on Physics, and A,C did get 5 on Calc.

A didn't get 5 on Calc. -- INCONSISTENT.

B is not the scholar.

............................

Suppose C is the scholar, then from her second statement:

C got 5 on calc, therefore did not get 5 on phys, and A,B did not get 5 on calc, and A,C did get 5 on calc.

A didn't get 5 on Calc. -- OK.
B didn't get 5 on Calc. -- OK.

This is the only case where all statements are consistent, so C is the scholar.