Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping/Linear, Numerical Distribution, Identify the Templates game.

This Grouping and Linear combination game features six players who must form four groups of two players each. The Linear aspect of this game is that the four games are played consecutively. However, one benefit for test takers is that no individual seat assignments are made within each game, and this limits the complexity of the game. The Grouping element results from the rules governing the formation of acceptable two-person games. From the game scenario it is immediately evident that some players will play multiple times, and you should actively seek a Numerical Distribution while working with the rules.

Our initial diagram references the six people, and represents the four games:

pt45_d04_g2_1.png

Rules #1 and #2. The first two rules can be added to the diagram easily. The first rule is reflected by T split-option on the second and fourth games (plus the Not Laws on the first and third games), and the second rule is represented by placing L in one of the spaces of the fourth game (it does not matter which space L occupies since the game does not assign seat positions):

• 
  pt45_d04_g2_2.png

Note that T could play in both the second and fourth games.

Rule #3. This rule establishes that N and P do not play in the same game, and it additionally establishes that N plays in one and only one game. The Grouping aspect of this rule can be diagrammed as:

pt45_d04_g2_3.png

We will discuss the numerical aspect of this rule during the discussion of rule #4.

Rule #4. With six players for eight spots, and with each player playing in at least one game, there are initially only two Numerical Distributions of the eight positions to the six players: 3-1-1-1-1- or 2-2-1-1-1-1. The first three rules do not provide us with any numerically useful information aside from the fact that T cannot be a player who plays in three games and that N must be a player who plays exactly one game, but the fourth rule is more useful because it eliminates one of the distributions. By establishing that S plays in exactly two games, this rule eliminates the possibility that a 3-1-1-1-1-1 distribution could exist. Hence, the eight positions must be distributed to the six players in a 2-2-1-1-1-1 relationship:

• 2: S (from the fourth rule, S plays in two games)
  2:
  1:
  1:
  1: O (from the third rule, O plays in one game)
  1: N (from the fourth rule, N plays in one game)

Consequently, the three players not assigned above—L, P, and T—must be distributed in a 2-1-1 relationship, where one of the three must play twice and the other two play just once.

The fourth rule also creates a powerful horizontal block:

pt45_d04_g2_4.png

Because this SOS block requires so much space, it can only be placed in two positions: games 1-2-3 or games 2-3-4. Because this block is limited, and because there are several other powerful rules, you should make the decision to diagram both of the templates created by the SOS block:

Template #1: SOS in games 1-2-3.

With the assignment of the SOS block to games 1-2-3, and with the placement of L in game 4 (from the second rule), we now have one player assigned to each game:

pt45_d04_g2_5.png

Because each player must play in at least one game, we must still assign N, P, and T to games. However, assigning N, P, and T would only fill three of the remaining four spaces, and so, from the distribution discussed previously, we know that the remaining space must be filled by one of L, P, or T (that is, one of those three people plays in two games).

At this point in the template, we have placed the SOS block and determined which players must still be assigned. Since there is only one open space in each game, we can ignore the NP not-block rule since N and P cannot now be placed together. We have also fully addressed the second and fourth rules, and so the only concerns still present in this template is which game(s) T will play in and who, aside from S, will be assigned to play twice (L, P, or T).


Template #2: SOS in games 2-3-4.

When the SOS block is assigned to games 2-3-4, the fourth game is now fully assigned to L and S. This placement affects T, who can only play in the second and fourth games. Thus, T must play in the second game:

pt45_d04_g2_6.png

The remaining three spaces must be occupied by N and P (who have thus far not been assigned to a game), and the apparent choice L, P, or T to play in a second game (someone must play twice and fill the final space). But, look closely at L, P, and T as possible two-game players: T can be eliminated because the remaining spaces are in the first and third games; P can be eliminated because selecting P would create the group of N, P, and P to fill in the final three spaces, and since two of the spaces are in the same game, the result would be that either N and P would play together (a violation of the third rule) or P would play both positions in one game (a violation of the stipulations in the game scenario). Consequently, we can conclude that L must be the player that plays two games, and that the group that fills the final three open spaces in this template is L, N, and P.

As if the determining the remaining players were not powerful enough, because two of the spaces are in the same game, we can also use the third rule to infer that one of N or P must play in the first game, and the other must play in the third game (if one of N and P does not play in the third game, then both play in the first game, a violation of the third rule). This forces L into the first game, resulting in a template with only two solutions:

pt45_d04_g2_7.png

The two templates give us an excellent overview of the general formations within this game, but because the first template has so many solutions, they do not give us complete information.