# Elimination Grid

Dean Peterson, Head of the Math Department at Peabody Polytech, is making the fall teaching schedule. Besides himself there are four other professors—Warren, Novak, Dornan, and Emerson. Their availability is subject to the following constraints.Warren cannot teach on Monday or Thursday.
Dornan cannot teach on Wednesday.
Emerson cannot teach on Monday or Friday.
Associate Professor Novak can teach at any time.
Dean Peterson cannot teach evening classes.
Warren can teach only evening classes.
Dean Peterson cannot teach on Wednesday if Novak teaches on Thursday, and Novak teaches on Thursday if Dean Peterson cannot teach on Wednesday.
At any given time there are always three classes being taught.

We indicate that a teacher does not work at a particular time by placing an X on the elimination grid. Placing the two conditions

*“Warren cannot teach on Monday or Thursday”*and*“Warren can teach only evening classes”*on the grid gives

Placing the remaining conditions in like manner gives

To answer the following questions, we will refer only to the grid, not the original problem.

Example-1

At which one of the following times can Warren, Dornan, and Emerson all be teaching?

- Monday morning
- Friday evening
- Tuesday evening
- Friday morning
- Wednesday morning

Solution

The grid clearly shows that all three can work on Tuesday night.
The answer is (C).

Example-2

For which day will the dean have to hire a part-time teacher?

- Monday
- Tuesday
- Wednesday
- Thursday
- Friday

Solution

Dornan and Novak are the only people who can work Monday evenings, and three classes are always in session, so extra help will be needed for Monday evenings.
The answer is (A).

Example-3

Which one of the following must be false?

- Dornan does not work on Tuesday.
- Emerson does not work on Tuesday morning.
- Peterson works on Tuesday.
- Novak works every day of the week except Wednesday.
- Dornan works every day of the week except Wednesday.

Solution

The condition
Hence, the answer is (D).

*“Dean Peterson cannot teach on Wednesday if Novak teaches on Thursday, and Novak teaches on Thursday if Dean Peterson cannot teach on Wednesday”*can be symbolized as**(P≠W)<—>(N=TH)**. Now, if Novak works every day of the week, except Wednesday, then in particular he works Thursday. So from the condition**(P≠W)<—>(N=TH)**, we know that Dean Peterson cannot work on Wednesday. But from the grid this leaves only Novak and Emerson to teach the three Wednesday morning classes.Example-4

If Novak does not work on Thursday, then which one of the following must be true?

- Peterson works Tuesday morning.
- Dornan works Tuesday morning.
- Emerson works on Tuesday.
- Peterson works on Wednesday.
- Warren works on Tuesday morning.

Solution

If you remember to think of an
gives
This tells us that Dean Peterson must work on Wednesday if Novak does not work on Thursday.
The answer, therefore, is (D).

*if-and-only-if*statement as an equality, then this will be an easy problem. Negating both sides of the condition**(P≠W)<—>(N=TH)****(P=W)<—>(N≠TH)****: Not all scheduling games lend themselves to an elimination grid. It’s sweet when this method can be applied because the answers typically can be read directly from the grid with little thought. Only one-third of the assignment games, however, can be solved this way. Most often the game will require a more functional diagram, and you will need to spend more time tinkering with it.**

Caution

Caution

When you first read an assignment game, you need to quickly decide whether or not to use an elimination grid. You may decide to use a grid. Then spend three minutes trying to set it up, only to realize you have taken the wrong path and have wasted three minutes. Unfortunately, exact criteria cannot be given for when to use an elimination grid. But this much can be said: if only two options (characteristics) are available to the elements—yes/no, on/off, etc.—then an elimination grid is probably indicated.

In the next game, which is considerably harder, more than two options are available to each element. It is, therefore, a game of multiple-choice.