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Setup and Rule Diagram Explanation

This is a Grouping: Defined-Fixed, Balanced game.

This type of game has shown up on a number of recent LSAT’s, and is now probably the game form most likely to show up on any given test. The reason, of course, is that this type of game has proven to be consistently difficult for the average test taker. This particular game features nine people filling nine total spaces in three groups. From a theoretical standpoint, there is an advantage in the fact that all nine people (or variables) must be used, since unused variables will not be a concern. As a lengthy aside, if some variables were unused, it may or may not be an advantage to the test taker. For example, if there were nine people for seven group positions, this would be advantageous since knocking out any two candidates would force the remaining seven people to fill the seven positions. However, if there were nine people for five positions, this would probably be disadvantageous since there are more options to fill each position. Knocking out one or two of the people wouldn’t necessarily produce any variables which would have to fill a position.

In this game, the nine variables themselves are split into two subgroups, children and adults, and you must make certain to keep this in mind throughout the game. The two groups can be represented as follows:
O94_Game_#2_setup_diagram 1.png
Any individual variable you work with can be symbolized with a C or A subscript (for example, QA) if necessary for clarity.

The first point of difficulty arises from the unbalanced separation of the adults. Since there are three groups and four adults, and each group must contain at least one adult, two groups contain a single adult and the remaining group contains two adults (a numerical distribution of 2-1-1, although not necessarily in that order). To partially combat this difficulty, you should reserve the bottom row of spaces for the adults that must be in each group, as follows:
O94_Game_#2_setup_diagram 2.png
It would not be wise to designate the top two rows as children-only since the remaining adult must be in one of those two rows. However, we can infer that each group contains either one or two children.

The rule involving the FJ block is clearly beneficial since both are children and therefore they will fill the two available child spots in any given group. Because of this fact, F and J cannot be in a group with any of the other children, an inference we can symbolize as follows:
O94_Game_#2_setup_diagram 3.png
The next two rules are similar in nature and can be represented with either not-blocks or double-not arrows. Both work efficiently, but for our purposes we will use not-blocks since this game has a linear-type setup conducive to using the visually powerful blocks:
O94_Game_#2_setup_diagram 4.png
In this case we have chosen to designate the adults with a subscript since they are the smaller group and thus somewhat more restricted.

The last rule provides us with two Not Laws on the second group:
O94_Game_#2_setup_diagram 5.png
This final rule also allows us to make some critical inferences. First, H and T have been reduced to
only two possibilities: group 1 or group 3. If for some reason one of those two groups should be completely filled by other variables, then both H and T must automatically go into the other group. As discussed previously, whenever variables are limited to only two options they are in a highly restricted state and will usually play a key role in the game. Also, if for some reason (as specified in a rule or “if” statement) another variable knocks either H or T out of group 1 or group 3, then H or T would have to go into the other group. In this game, the third and fourth rules act in such a way. According to the third rule G and T cannot be in the same group. Since T cannot be in group 2, we can make the following inferences:
O94_Game_#2_setup_diagram 6.png
Taking the contrapositive of the first inference, if T is not in group 3, then G is not in group 1. This is identical to saying that if T is in group 1, then G is in either group 2 or 3. The same reasoning applies to the second inference. These contrapositives, while helpful, simply reflect the not-block information in the rules and in fact in this case do not produce any truly helpful information. However, it is always of value to quickly examine the contrapositive of any conditional rule you see in a game. Some of the most important game insights have come from understanding the implications of the contrapositive.

The last rule also interacts with the fourth rule to produce the following inferences:
O94_Game_#2_setup_diagram 7.png
The same contrapositive reasoning as above applies to this set of inferences as well.

We have now exhausted the possible inferences from the initial set of rules and we should be ready to attack the questions. In approaching the questions, rely upon the restricted variables F, J, R, H, G,
and T to help make deductions essential to answering the questions. A close examination of the restricted points in any game—whether variables or spaces—almost always yields some inferences of value.
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Hi there,

Thank you for posting this! I was wondering if you could suggest other similar problems to practice with?

Thank you!
 Christen Hammock
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Hi Ck!

The Logic Games Bible and Bible Workbook both have practice problems specific to game types. Either one would provide plenty of practice for Defined/Balanced Grouping games!
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Thanks so much! I'll look for some in there :)
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How do we know to distribute the variables evenly among each group? Why can’t there be, for example, 4 people in one group and 3 and 3 in the others. Basically how do I know that the nine variables will be spread out evenly among the 3 different groups if the rules do not specify it
 Luke Haqq
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Hi gabmarr!

You ask,
Basically how do I know that the nine variables will be spread out evenly among the 3 different groups if the rules do not specify it
However, it seems that the rules do in fact specify this. The instructions for the game explain, "The canoeists will be divided into three groups -- groups 1, 2, and 3 -- of three canoeists each." We therefore know that the nine variables are going to be split up into three groups of three. This instruction wasn't with the list of rules and conditions but rather was in the initial paragraph of instructions for the game, which might have made it difficult to spot.
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I was wondering how do we assume that each canoe can only accommodate 2 kids maximum? Which would enable the FJ block to create the inference that HGK must be placed on different canoes. Why cant it be a 3-1-1 distribution with the kids? Such as Canoe 1 having QFJG?
 Rachael Wilkenfeld
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Hi lranderson,

The scenario tells us that the canoes only fit three people each, and the first rule says that every canoe needs an adult. That only leaves a maximum of two slots per canoe for kids. I'm guessing you overlooked the canoe size from the scenario, which is very easy to do. It's tempting to jump right into the rules in a game, but reading that little paragraph above matters too.

Hope that helps!

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