$\textbf{Class Substitute}$
Kurt, a math professor, needs a substitute for one of his classes today. He sends an email to his three closest co-workers: Julia, Michael, and Mary asking if anyone can help. However, Prof Kurt forgets to give the details of his class. Julia, the department chair, knows which class Kurt teaches, but does not know the time nor the building. Michael plays racquetball with Kurt often, so he knows what time Kurt teaches, but does not know other details. Mary happens to know which building Kurt's class is in, but neither the class itself nor the time.The possible candidates for Prof Kurt's class are list below.
- 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 over the list, 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 now know." Which class does Kurt need a substitute for?
$\textbf{Answer}$
It is Calc $2$ at $10$ in North Hall.
$\textbf{Analysis}$
One effective way to solve this type of problem is to list all the conditions in a table and then work from there.
$$\begin{array}{l|c|c|c|c|c} & 9 & 10 & 11 & noon & 3\\ \hline East\ Hall & & 1 & 2 & & \\ \hline South\ Hall & & 1 & & 2 & \\ \hline West\ Hall & & & & 2, 3 & 1 \\ \hline North\ Hall & 1 & 1, 2 & & & \\ \hline\end{array}$$
Julia will know the answer for sure if she can uniquely identify the class. In this table, only Calc $3$ appears once. By the fact that the other two know for sure that Julia does not know the answer, the class cannot be Calc $3$ which means the class cannot be at noon in West Hall. Let's blank out this row and column.
$$\begin{array}{l|c|c|c|c|c} & 9 & 10 & 11 & noon & 3\\ \hline East\ Hall & & 1 & 2 & & \\ \hline South\ Hall & & 1 & & & \\ \hline West\ Hall & & & & & \\ \hline North\ Hall & 1 & 1, 2 & & & \\ \hline\end{array}$$
Since Michael cannot be sure, the time cannot be $9$ nor $11$. Similarly, since Mary cannot be sure either, the location cannot be in the South Hall. Blanking out these will leave the possibilities as follow:
$$\begin{array}{l|c|c|c|c|c} & 9 & 10 & 11 & noon & 3\\ \hline East\ Hall & & 1 & & & \\ \hline South\ Hall & & & & & \\ \hline West\ Hall & & & & & \\ \hline North\ Hall & & 1, 2 & & & \\ \hline\end{array}$$
There are two Calc $1$ classes and one Calc $2$ class left. For Julia to be certain, the class must be the Calc $2$. Done.