Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping/Linear Combination game.

Since the concertos from each composer produce the variables that are used in the linear setup, they are the logical starting point for our analysis. Each composer supplies two concertos, and the groups are as follows:

F94_Game_#4_setup_diagram 1.png

Of the three groups, Giuliani and Vivaldi are the most restricted, since they only have three concertos to fill the required two selections per composer. Thus, if any one concerto is unavailable from either the Giuliani or Vivaldi group, then the other two concertos must be selected. In a situation such as this, it is always best to immediately check the rules for any negative grouping rules among the members of the restricted groups. The third rule contains such a relationship:

• X Z

Since X and Z can never be selected together, we can Hurdle the Uncertainty and infer that Y must be selected from Vivaldi’s group:

F94_Game_#4_setup_diagram 2.png

The scenario above, three variables for two spaces, is perhaps the most common inference scenario that appears in Grouping games. Any negative grouping rule or any question stem that knocks out one of the three variables leads to the inference that some other variable must be selected. In the above scenario, the rule involving X and Z effectively knocks one of those two variables out of the selection pool, forcing Y to be selected. One of the best examples of this type of inference occurred in a game from the 1980s. In that game, seven basketball players were selected for five starting spots. Clearly, this leaves only two extra variables in the selection pool. However, as the rules unfold it turns out that two separate pairs of variables could not be selected together, in each case effectively reducing the candidate pool by one player. Since this occurred twice, it had to be that the three players not involved in the negative grouping rules were selected, a classic Hurdle the Uncertainties situation:

F94_Game_#4_setup_diagram 3.png

Of course, a similar scenario can be produced with a wide variety of numerical combinations, four candidates for three spaces, eight candidates for six spaces, etc. It is also important to note that many questions introduce “if” statements that ultimately result in limited scenarios, such as three candidates for two spaces or four candidates for three spaces. The point is that any selection group that is limited in size relative to the number of members that must be selected will probably yield an important inference, and you must always watch for situations such as these in games. In the guitar concerto game under consideration, in Question #23 we benefit directly from our inference that Y must always be selected. Continuing with the setup of the game, we arrive at the following representation of the rules:

F94_Game_#4_setup_diagram 4.png

A combination of the first and second rules produces the following additional deduction:

• N M

This deduction provides the answer to Question #24. There are also several other, less important inferences that can be made. For example, according to the second rule, when M is selected, J and O cannot be selected. Via the contrapositive, when J is not selected then N cannot be selected. From Rodrigo’s group then, when M is selected, P is also selected:

• M P

And since J is not selected, H and K must be selected from Giuliani’s group:

• M P, H, K

However, when P is selected, then X is not selected, and thus from Vivaldi’s group Y and Z must be selected:

• M P, H, K, Y, Z

So, if M is selected, the other five positions are automatically filled. A similar situation arises with X. When X is selected, Z and P are not selected. Since P is not selected, M cannot be selected (see the inference above). Since both M and P are not selected, N and O must be selected from Rodrigo’s group. Of course, Y is always selected:

• X Y, N, O

According to the first rule, if N is selected, then J must be selected:

• X Y, N, O, J

These last two major inferences involving M and X are helpful, but they are not essential to answering the questions in the game. We discuss them here simply to indicate the type of inferences that can follow from restricted situations. The only other rule of note is the last rule, which states that if X is played on the fifth Sunday then one of Rodrigo’s concertos must be played on the first Sunday. This rule is noteworthy because it is so specific. It should be easy to track while answering the questions, because it relies so heavily on two designated spaces.

[scyq6@sina.com]

I think this is a grouping game. I have trouble understanding question 23 and 24, the last two. I might miss a key inference from the setup.

For 23, how do I make the inference from the setup to determine which concerto has to be selected? I can only eliminate "J" from the previous question.

Thank you!

[Emily Haney-Caron]

Thanks so much for the question!

With 23, I think you're right that you're missing an inference. Make sure as you look over your set up that you're keeping in mind that we have to have exactly 2 concertos by each composer. Now look at the rule that if X is selected, Z can't be. X, Y, and Z are the only 3 concertos by Vivaldi. Since X and Z can't be in together, we know that one of them will be in, and Y will HAVE to be in, every single time.

24 is also about making inferences from your initial set-up. When you combine the first and second rules, we know that if M is selected, J can't be selected, AND N can't be selected. If we take the contrapositive of that, if J is selected, M can't be selected, and if N is selected, M can't be selected. So, based on that set-up, I know what the answer to 24 has to be.

This is definitely an inference-heavy game; spend some more time looking it over and maybe try the set-up again, and let us know if you have more questions!

[Tamirra]

I set this up erroneously using 1 2 3 4 5 6 as the base and then listed the composers and the songs on the side. It seemed like the Sundays were the most stable.

Is there a way to avoid this? Something to look for?

I fall into the "linear must mean a line of numbers" trap a lot with these Grouping/Linear games.

Thanks!
Tamirra

[Rachael Wilkenfeld]

Hi Tamirra,

Good question. This is a grouping/linear combination game so our first task is to figure out the grouping aspect. Grouping is always primary in a grouping/linear combination game. You can't put variables in order until you know what variables you are working with. In this case, we are even more focused on the grouping aspect, because almost all of the rules themselves relate to the grouping relationships. The ideas of who can/can't be selected together really drive the grouping/linear games.

The linear aspect of the game gets filled in later---once we know the variables selected, we can think about what order they go in. However, in this game, we only have two real limitations driving the ordering of the variables. If J and O are both selected, J is before O. And if X is in 5, M/N/O is in 1. We'd want to only worry about those rules once we've met the sufficient conditions.

Hope that helps!
Rachael

[lsathelpwanted]

I just re-did this problem from long ago.

I solved it by writing the rules and contrapositives. For my inferences, I made a "tree" type diagram and I found most of the key inferences. I did not recycle my rules and inferences to create the list of inferences PS has made. And that worries me!

Should I only use tree diagrams on pure sequencing games? I think I may have confused the two approaches.

[Jeremy Press]

Hi lsathelpwanted,

The approach in the original setup here is very sound, and it'll get you to the inferences you need in this game. We like this approach for most students initially, because there's a tendency among students to either "misread" or "miscreate" (i.e. put the chain together inaccurately) long chain-type diagrams in grouping games; or to feel like something was left out if they don't reproduce the contrapositive of the long chain they create.

But, for students who are very comfortable with conditional reasoning, and who aren't likely to make mistakes in reading chains, the chain approach you describe can be a very effective and very simple one. I'd have to see your chain diagram to confirm its accuracy, but assuming it was accurate there's nothing wrong with approaching the game this way. There's often more than one acceptable way to approach the setup of a complex game like this. Assuming you were accurate when you answered the questions, and assuming you didn't take an overly long time to complete the game, go with what works! One caution in this game particularly: you can't actually link all the conditional rules here together (which is something you can sometimes do on undefined grouping games). So if you did that, I'd encourage you to go back and see if you made a mistake in your chain construction.

Another caution: I just realized by "tree" you might mean a sequencing map like you'd find in a pure sequencing game. Just like there's not one single conditional chain that includes all the variables in this game, there is also not one single sequencing map that includes all the variables. So if that's what you arrived at, take a look back and see if you missed some of the openness in this game.

I hope this helps!