Six park rangers, J, K, L, M, O, and P, are each to be assigned to monitor one of three areas, area 1, area 2, and area 3, in a national park. At least one ranger, but not more than three, is assigned to each area. The assignment must conform to the following conditions. M is assigned to area 3, neither O nor P is assigned to area 1, L is assigned to the same area as either K or M, but not to the same area as both.

If O is assigned to area 2, then J is assigned to the same area as K. Otherwise, J is assigned to a different area than K. So we have six park rangers, and we are trying to assign them to three different areas. We are not concerned with order here, so this is not a linear game. We are dealing with a grouping game here.

We're just trying to figure out which rangers can go into which group. So that's our primary focus. So let's make sure we have a good grasp of the variables. We have our rangers, J, K, L, M, O, and P. And they're gonna be distributed into three areas, 1, 2, and 3. Now we have some minimum and maximums here.

Each area's gonna have at least one ranger. And each area can have a maximum of three rangers. So we can't have more than three rangers in a particular area. And each area has to have at least one ranger, and the passage also tells us that every ranger is assigned. So we're clear on how our variables are going to interact with the diagram, or the rangers are going to interact with the areas.

So if we sketch out a diagram, this is what it would look like. So even though we're not dealing with a linear game, we can still sort of chart out our groups here. So we have area 1, 2, and 3, and then our variable set that will be distributed into that. Now, let's look at our rules.

Rule number 1, M is assigned to area 3. Always really great rules when they just tell you where something goes. So we can go ahead and just plug that into our diagram. Rule number 2, neither O nor P is assigned to area 1. Also great rules when they tell you what can't go somewhere. So we're not gonna have P or O going in area 1.

Rule number 3, L is assigned to the same area as K or N, but not both. So when we look at our final arrangement, we're gonna have, in one of the areas, either L or K, or either L or M, or L or K. But we will never have all those variables together. So we can never have an instance where we have L, M, and K in the same area. So L pairs with one or the other, but not both.

Then rule number 4 says, if Olsen is assigned to area 2, then J is assigned to the same area as K. Otherwise, so when O is not assigned to area 2, J is assigned to a different area than K. So this is a conditional here. Whenever we have O in 2, J and K are going to be assigned to the same area.

Now, this means the same area as each other, not the same area as O. So this does not mean that we're gonna have O in 2 and then J and K in 2. It means that if O is in 2, wherever J and K go, they need to go to the same area together. Now, if O is not assigned to 2, so in theory, that would be being assigned to 1 and 3, but we have a rule that already tells us that O cannot be assigned to 1.

So if O is not assigned to 2, that means that O is assigned to 3. In that instance, J and K cannot be in the same area. So if O is in 3, then J and K must be sort of spread out between the other areas, but both those variables cannot be in the same area. So understanding the conditional nature of these rules and sort of the implications of the JK sequence, or the JK sort of occurrence block that happens here, is important.

All right, so now we have our master diagram, there are not many inferences that we can get here. There's not much that we can really plug into our diagram here, so this is what we're going to end up with. It's good information to help us navigate through the questions.