June 2007, Game 4, Setup

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Game four. The scenario here indicates that we have a floating grouping game. There are three groups, the recycling center is one, two, and three and we have five things to split up between them. Each center has to do at least two and no more than three. So already from the beginning, we know that we have to use at least one of our items more than once, because if there are three centers and two spaces a center, that's six spaces total.

And we could have more because there could be up to nine spaces if all three centers did three things. The information that we have so far is enough for us to build the basic framework of slots. Make three groups, put two slots into each because that's all you know is going to be in each, and put a note off to the side that says that you have a maximum of three slots per center.

And of course, put your roster down of g, n, p, t, and w. Now proceed on to the rules. The first rule is a conditional rule. So write both it and its contrapositive. If a center does w it has to do n as well. Which means that if it doesn't do n, it won't be able to do w.

The best place to put that is in short hand off the side because it won't affect the spaces directly. The next rule is also a conditional rule but this time it's about the slots in the centers. Basically anything that center 2 does, center 1 also has to do. So with this rule, write it and its contrapositive.

So if it's in 2, it's definitely in 1. If it's not in 1, it is not allowed to be in 2. Be careful, this rule does allow for center 1 to do things at center 2 doesn't. But since center 2 is going to do two things, center 1 will have to do those same two things. So if center 1 is going to do something that center 2 doesn't, it would have to get an extra slot.

There have to be three slots and one for it to do something that center 2 doesn't. Rule three. So rule three is a two part rule. The first part says that only one recycling center recycles plastic. So note that over the plastic that you only get one of those. Now this would allow for an immediate deduction.

If you only get one plastic that means that center 2 can't do plastic because if center 2 were doing plastic, center 1 would have to do it as well and that would be two plastics. Now the second half of the rule is an if then statement. So if plastic is recycled somewhere, glass can't be. The contrapositive of that would be if glass is recycled, then plastic can't be.

You can't do both glass and plastic but you could do neither glass nor plastic. If these conditional rules and forming the contrapositive is a little rusty or shaky, take a look at the lessons that we have elsewhere in the course on conditional reasoning. Now there aren't a lot of deductions that we can make beyond the deduction about p not being in 2.

And if you look over at the questions, you can see that most of them are if questions, so local questions that are gonna introduce a new condition and require a new sketch. So we probably don't need to do anything else with the master sketch. This will be enough to carry us on. So now on to the questions.

The best question order here is probably to do 18 first, skip 19, go to 20, 21, 22, and 23. And double back for 19. That is you always wanna do your solution questions first then move on to the local questions, then double back to get the global questions or other weird types of questions.

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