Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping: Defined-Fixed, Unbalanced: Underfunded game.

This game establishes two groups—G and L—that are planted with trees from a group of four varieties—M, O, S, and T. The game scenario does not establish how many trees are in each park, but fortunately the first rule fixes the number in each park at three. Because there are only four varieties to choose from, we can immediately infer that there must be some overlap and that some varieties are planted in both parks.

The initial setup:

PT56-Dec2008_LGE-G3_srd1.png

Rule #2. This rule establishes that an MS block must appear in at least one of the parks:

PT56-Dec2008_LGE-G3_srd2.png

Rule #3. This rule creates a simple conditional relationship:

PT56-Dec2008_LGE-G3_srd3.png

Note that when O is planted in a park, then T must be planted in that same park, leaving only one remaining space. Thus, when O is planted in a park, the MS block created in the second rule cannot be in the same park, and must be in the other park. This also means that O cannot be planted in both parks, an inference tested in question #15.

However, there is also another inference that can be drawn from this rule, one based on the contrapositive. The contrapositive of the rule is:

PT56-Dec2008_LGE-G3_srd4.png

If T is not planted in a park, then O cannot be planted in that same park, leaving only M and S to be planted in the park. However, because the first rule requires three varieties to be planted in each park, this is an unacceptable situation. Thus, we can infer that T must be planted in each park:

PT56-Dec2008_LGE-G3_srd5.png

This is clearly a powerful inference, and one that is tested directly in question #13.

Rule #4. This rule adds M to G:

PT56-Dec2008_LGE-G3_srd6.png

With M planted in G, we know that if S is planted in G then the second rule will be satisfied; otherwise, if S is not planted in G, it must then be planted in L along with M. In either event, S must be planted in at least one of the two parks (and possibly both):

PT56-Dec2008_LGE-G3_srd7.png

Note that neither the scenario nor the rules indicates that each of the four varieties must be planted. It is therefore possible that one of the varieties is not planted in either park. This leaves only two possible numerical distributions of the six planting slots to the four trees:

PT56-Dec2008_LGE-G3_srd8.png

Note that a 3-1-1-1 distribution and other variations featuring 3 or more are impossible because the most a tree can be planted is twice (as there are only two parks).

In the 2-2-2-0 distribution, because O cannot be planted in both parks, O is not planted at all, and the other three varieties are each planted twice:

PT56-Dec2008_LGE-G3_srd9.png

In the 2-2-1-1 distribution, T must be planted twice, and because O cannot be planted twice, it must be planted once. M and S then fill the remaining options:

PT56-Dec2008_LGE-G3_srd10.png

Combining all of the above information leads to the final setup for the game:

PT56-Dec2008_LGE-G3_srd11.png

The key in this game is to watch the MS block, which requires tracking both the one open space in G and the planting of S.

[Dave Killoran]

This is a Grouping: Defined-Fixed, Unbalanced: Underfunded game.

This game establishes two groups—G and L—that are planted with trees from a group of four varieties—M, O, S, and T. The game scenario does not establish how many trees are in each park, but fortunately the first rule fixes the number in each park at three. Because there are only four varieties to choose from, we can immediately infer that there must be some overlap and that some varieties are planted in both parks.

Rule #1 establishes that each park has exactly three varieties:

• ___ .....___
  ___ .....___
  ___ .....___
   G L

Rule #2. This rule establishes that an MS block must appear in at least one of the parks.

Rule #3. This rule creates a simple conditional relationship:


O T

Note that when O is planted in a park, then T must be planted in that same park, leaving only one remaining space. Thus, when O is planted in a park, the MS block created in the second rule cannot be in the same park, and must be in the other park. This also means that O cannot be planted in both parks, an inference tested in question #15.

However, there is also another inference that can be drawn from this rule, one based on the contrapositive. The contrapositive of the rule is:

T O

If T is not planted in a park, then O cannot be planted in that same park, leaving only M and S to be planted in the park. However, because the first rule requires three varieties to be planted in each park, this is an unacceptable situation. Thus, we can infer that T must be planted in each park:

• ___ .....___
  ___ .....___
   T .....T
  G L

This is clearly a powerful inference, and one that is tested directly in question #13.

Rule #4. This rule adds M to G:

• ___ .....___
   M .....___
   T .....T
  G L

With M planted in G, we know that if S is planted in G then the second rule will be satisfied; otherwise, if S is not planted in G, it must then be planted in L along with M. In either event, S must be planted in at least one of the two parks (and possibly both):

• S/ .....___
   M /S
   T .....T
  G L

Note that neither the scenario nor the rules indicates that each of the four varieties must be planted. It is therefore possible that one of the varieties is not planted in either park. This leaves only two possible numerical distributions of the six planting slots to the four trees:

• 2-2-2-0 (three trees are planted twice, one tree is not planted)
  or
  2-2-1-1 (two trees are planted twice, two tree are planted once)

Note that a 3-1-1-1 distribution and other variations featuring 3 or more are impossible because the most a tree can be planted is twice (as there are only two parks).

In the 2-2-2-0 distribution, because O cannot be planted in both parks, O is not planted at all, and the other three varieties are each planted twice:

• 2  2 2 0
  M S T O

In the 2-2-1-1 distribution, T must be planted twice, and because O cannot be planted twice, it must be planted once. M and S then fill the remaining options:

• 2 2 1   1
  T M/S   S/M O

The key in this game is to watch the MS block, which requires tracking both the one open space in G and the planting of S.

[Tommy2456]

Hello, I am doing the North American Tree Planting question, I am not sure of the specific game it came from. But for the setup, why is it that you need to have three setups instead of two? In the rules it states that at least one of the parks is planted with both maples and sycamores, NOT 3. How do you know when to use one or more setup boards?

[Nikki Siclunov]

Tommy,

Thanks for your question. The game is from the December 2008 LSAT.

The reason why we have three templates for this game is because the MS-block can either be assigned to 1) Graystone only; 2) Landing only; or 3) both Graystone and Landing. So, there are three possibilities for the placement of the MS-block. The remaining inferences in each solution follow from the application of the other rules.

An alternative approach is to use only two Templates: one in which MS is assigned to Graystone, the other - where MS is assigned to Landing, keeping in mind that in each template the block can also be assigned to the other park. Either way, the key is to recognize that T must be used in both parks, because without T, we cannot have O (contrapositive of the third rule), making it impossible to have 3 different varieties of trees per park.

Hope this helps!

[Tommy2456]

Ahhhh!! It makes perfect sense now? So at least doesn't necessarily mean that it has to be applied to 1 park, that could mean both?

[Robert Carroll]

Tommy,

If it's consistent with the rules that each park has M and S, then nothing about "at least" excludes that possibility. "At least" sets a minimum rather than a maximum.

Robert Carroll