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June 2007, Game 4, Setup


Game 4, the scenario here indicates that we have a floating grouping game. There are three groups, the recycling centers 1, 2, and 3 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 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 gnpt and w. Now, proceed on to the rules. The first rule is a conditional rule. So, right both it and its contrapositive. If a centre 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 shorthand 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 and the centers, basically anything that center 2 does, center 1 also has to do. With this rule, right it, and it's contrapositive. So, if it's in two, it's definitely in one.

If it's not in one, it is not allowed to be in two and be careful. This rule does allow for center 1 to do things that 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'd have to be three slots and one for it to do something that center 2 doesn't. Rule three. So, rule three is a three 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 will allow for an immediate deduction if you only get one plastic that means that centre 2 can't do plastic.

Because of center 2, we're doing plastics, 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 into. And if you look over 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. Remember, you don't have to do the questions in the order that they are given to you, can skip around a bit. And here it would probably be best to start with question 18 because it's one of those global questions that asked for one complete possible outcome of the game and you can do those with just rules.

Then, skip on to the local questions that give you a rule. So, that would be 20, 21, 22 and 23. And then, double back to pick up that global question that's left, question 19.

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