# Circuit

Six lights—J, K, L, M, N, O—are connected in a circuit. Each light can be either on or off.

If K is on, L is off.
J and N cannot both be on.
M is off if and only if either J or N is on.
If O is on, N is on; if O is off, N is off.

To keep the notation simple, let the letter itself stand for

*“the light is on,”*and place a tilde before the letter to indicate that the light is off. The condition

*“If K is on, L is off”*is naturally symbolized as

**K—>~L.**The condition

*“J and N cannot both be on”*means that if one is on the other must be off:

**N—>~J.***The condition

*“M is off if and only if either J or N is on”*is naturally symbolized as

**~M<—>(J or N)**. Finally, the condition

*“If O is on, N is on; if O is off, N is off”*means that O is on if and only if N is on:

**O<—>N**.

Now we come to the crucial decision—with which condition should we start our chart? Following the guidelines on page 139, look for the element that occurs in the greatest number of conditions; it is N. Of the three conditions that contain N, the condition

**O<—>N**is the most restrictive, so we start the flow with it:

**O<—>N**

Next, adding the condition

**N—>~J**gives

**O<—>N—>~J**

Then, adding the condition

**~M<—>(J or N)**gives

Finally, the condition

**K—>~L**is independent of the other conditions, so the flow chart consists of two distinct parts:We will use only this chart to answer the following questions.

Example-1

How many lights in the circuit must be on?

- 0
- 1
- 2
- 3
- 4

Solution

Since this is a counting problem, we anticipate that it will be hard. It won’t disappoint us.

From the bottom chart, we see that both K and L can be off. (The condition
Next, from the top chart if both O and N are on, then both J and M must be off. Thus it is possible for only two lights, O and N, to be on, which eliminates (D) and (E).
Next, we check the alternative circumstance where both O and N are off. From the top chart, we see that if M is also off, then J must be on. While if M is on, J can not be on. Combining these cases (M
Hence, it is possible for only one element to be on, which eliminates (C).
Finally, we check whether all the lights could be off.
Again, the top chart shows that this is not possible: if M is off, then either N or J must be on, which eliminates (A)—at least one light must be on.
The answer, therefore, is (B).

From the bottom chart, we see that both K and L can be off. (The condition

**K—>~L**states only that if K is on then L is off. It says nothing about the case when K is off, so both K and L*could be*off.) Furthermore, since the bottom chart is independent of the top chart, the status of K and L (whether on or off) does not affect the other lights.*on*or*off*), we see that one and only one of M, J, and N must be on.Example-2

If J is on, which one of the following could be a complete and accurate list of the other lights that are on as well?

- K and O
- L and N
- L and O
- M, N, and K
- L

Solution

Notice that (A), (B), (C), and (D) all contain either N or O, but not both.
However, the condition
This eliminates (A) through (D).

**O<—>N**states that they are always on at the same time.