Six entertainers— Robinson, Shahpari, Tigay, Wu, Yeaton, and Zane—are being scheduled for the six performances on the opening day of a community festival. Each entertainer will perform at one of six times—in the morning at 9:00 A.M., 10:00 A.M., or 11:00 A.M., or in the afternoon at 2:00 P.M., 3:00 P.M., or 4:00 P.M.—with no two entertainers performing at the same scheduled time. The order in which the entertainers perform is subject to the following constraints:”
We have somewhat of a strange setup here. It's a linear game, but notice that while obviously in sequential order, 9 AM, 10 AM, 11 AM, 2 PM, 3 PM, 4 PM, we obviously have a gap there between 11 AM and 2 PM and we obviously need to distinguish between the morning and the afternoon.
In terms of our base, we're going to set it up as follows. We're going to have AM, our three positions, 9, 10, and 11 and then we are going to have our PM, which is going to be 2, 3, or 4. We have six variables that we are going to place. Those variables again are going to be R, S, T, W, Y, and Z. You notice that no two entertainers are going to perform at the same scheduled time, so this game is one-to-one. Each of these entertainers is going to show up once and only once.
So with that in mind, let's turn our attention to the constraints:
The first rule tells us that, “Robinson must perform as some time before Zane.” We know that Robinson performs before Zane. We know that Zane cannot be at 9 AM and that Robinson cannot be at 4 PM.
Our second rule states that, “Yeaton's performance must be the next performance after Wu's.” We know that we're going to have Wu followed by Yeaton. We have this block that is now established. Notice this block tells us that Y cannot be first and similarly that W cannot be last.
It’s important to note here that nothing in this rule prevents W from appearing at 11 AM and Y from appearing at 2 PM. All condition 2 tells us is that Y’s performance must be the next performance after W’s and since 2 PM is the next performance after 11 AM that rule would be satisfied if W was in 11 and Y was in 2.
The next rule states that, “Tigay must perform in the afternoon.” We know that if we are Tigay, we are going to appear in the PM. Contrapositive tells us that if it's the AM, we will not have Tigay appear, so T will not be appearing in the morning but T will be appearing in the afternoon at some time. So we'll put T over here.
And the final constraint tells us that, “Zane must perform in the morning.” Again, we know that if Zane, we are appearing in the AM. Contrapositive, if we are in the PM, it is not Zane. Again, we know that Z can't be at 2, 3, or 4. Again, we know that Z must appear in the morning, so we'll place Z over here to keep that clear.
A couple of things we should note at this point. For 9 AM, we've eliminated all but three of the variables. Z, Y, and T cannot appear at 9 AM, which means it has to be R, S, or W. Similarly, we've eliminated three variables from the 4 PM spot, R, W, and Z, which means that it must be S, T, or Y.
Another very important thing to note is that we have eliminated Z from all but two possibilities. Z can only appear 10 or 11. Given that that is very restrictive, we're going to want to draw out the possibilities.
Turning our attention here to the right, let's try out these different scenarios. First, I'm going to place Z in 10 AM. The first rule tells us that R must come before Z, so that means R must be 9 AM. Now, we know that Y cannot appear before W, so Y cannot be in the 11 slot, which means now the only possibilities remaining here for that position because we know again T can't be in the morning, so T is eliminated from 11 AM. The only two possibilities here are S and W, which tells me that in our afternoon positions, we're going to have Y, T, and the reverse of W and S. You notice placing Z in 10 AM gives us a very nice scenario here. Obviously we can't draw it out completely, but we do have a nice sense of what's taking place.
Now let's turn our attention to the next possibility, which would be Z in 11 AM. Again, we know T cannot be in the morning. Notice now what's going to take place here. We have a couple possibilities. If we have our WY block, the only possibility for that would be 9, 10, but the problem with that is where would R go. Correct? That tells me that W cannot appear in the morning, which also tells me that Y cannot appear in the morning. That leads us to a huge deduction here. W cannot be in 9 or 10. Y cannot be in 9 or 10, which means that W and Y must appear in the afternoon, either 2, 3 or 3, 4. You notice what those have in common is that no matter what, 3 is going to be W or Y.
We also know now that one of those other positions is going to be the reverse of W and Y and then T. What we have here is T and then the opposite of W and Y left to place. That tells me that the remaining variables R and S must appear in the morning. The order is not relevant. We just know that R has to come before Z. That is going to occur in either scenario here.
That would be the setup for this game. Again notice the key here. The key here for us was Z was extremely restrictive. It could only appear in two positions, so we tried them out. We had scenario one where Z is 10 AM and scenario two where Z is 11 AM. That led to a huge deduction here that Y must appear in the afternoon. We will write that out here next to our main setup. Y also must appear in the afternoon. Y cannot appear in the morning sessions.
Now that we have our setup, our deductions, let's turn our attention to the questions.