The Substitute - Futility Closet

Between July 2015 and October 2018, the National Security Agency offered a monthly puzzle written by a staff member. Here’s the puzzle for October 2015, created by applied research mathematician Ben E.:

Kurt, a math professor, has to leave for a conference. At the airport, he realizes he forgot to find a substitute for the class he was teaching today! Before shutting his computer off for the flight, he sends an email: “Can one of you cover my class today? I’ll bake a pie for whomever can do it.” He sends the email to Julia, Michael, and Mary Ellen, his three closest friends in the math department, and boards the plane.

As Kurt is well-known for his delicious pies, Julia, Michael, and Mary Ellen are each eager to substitute for him. Julia, as department chair, knows which class Kurt had to teach, but she doesn’t know the time or building. Michael plays racquetball with Kurt so he knows what time Kurt teaches, but not the class or building. Mary Ellen helped Kurt secure a special projector for his class, so she knows what building Kurt’s class is in, but not the actual class or the time.

Julia, Michael, and Mary Ellen get together to figure out which class it is, and they all agree that the first person to figure out which class it is gets to teach it (and get Kurt’s pie). Unfortunately the college’s servers are down, so Julia brings a master list of all math classes taught that day. After crossing off each of their own classes, they are left with the following possibilities:

Calc 1 at 9 in North Hall
Calc 2 at noon in West Hall
Calc 1 at 3 in West Hall
Calc 1 at 10 in East Hall
Calc 2 at 10 in North Hall
Calc 1 at 10 in South Hall
Calc 1 at 10 in North Hall
Calc 2 at 11 in East Hall
Calc 3 at noon in West Hall
Calc 2 at noon in South Hall

After looking the list over, Julia says, “Does anyone know which class it is?” Michael and Mary Ellen immediately respond, “Well, you don’t.” Julia asks, “Do you?” Michael and Mary Ellen both shake their heads. Julia then smiles and says, “I do now. I hope he bakes me a chocolate peanut butter pie.”

Which class does Kurt need a substitute for?

	Select Click for Answer
1) Since Julia only knows the class name, the only way she could immediately know is if it was Calc 3. Since Michael and Mary Ellen both know that Julia doesn’t know, that means they know the class isn’t Calc 3. Since Michael only knows the time, that means the class can’t be at noon. Because Mary Ellen only knows the building, that means the building can’t be West Hall. That leaves only the following possibilities: Calc 1 at 9 in North Hall Calc 1 at 10 in East Hall Calc 2 at 10 in North Hall Calc 1 at 10 in South Hall Calc 1 at 10 in North Hall Calc 2 at 11 in East Hall 2) Since Michael doesn’t know which class it is, that means the time can’t be 9 or 11. Since Mary Ellen doesn’t know either, the class can’t be in South Hall. That leaves only three possibilities: Calc 1 at 10 in East Hall Calc 2 at 10 in North Hall Calc 1 at 10 in North Hall 3) At this point, Julia now says she knows the answer. Since there are two Calc 1 classes, it must be that Julia knows the class is Calc 2. Thus, the class is Calc 2 at 10 in North Hall.

Click for Answer