Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is an Advanced Linear: Balanced, Identify the Templates game.

The game scenario and rules combine to form the following main diagram:

J02_Game_#4_setup_diagram 1.png

The game scenario indicates that laps 1 and 6; 2 and 7; 3 and 8; 4 and 9; and 5 and 10 are all paired, so that a member who swims one lap in the pair automatically swims the other lap in the pair. For example, if a member swims lap 1, he or she also swims lap 6, and if a member swims lap 6, he or she also swims lap 1. Consequently, if a member cannot swim one of the laps in a pair, he or she cannot swim the other lap in the pair. For example, the rules state that J cannot swim lap 9. By deduction, J also cannot swim lap 4.

The pairing of the laps also has an unusual effect on the last rule. The last rule is rather carefully worded to say, “At least one of J’s laps is immediately after one of O’s laps” (italics added). In a regular linear game, a rule like this would normally create an OJ block where J cannot swim lap 1 and O cannot swim lap 5. In this game, however, because laps 5 and 6 are consecutive, J can swim lap 1 and O can swim lap 5, as in the following hypothetical:

J02_Game_#4_setup_diagram 2.png

In the above hypothetical, J in lap 6 swims immediately after O in lap 5, meeting the specification in the last rule.

In the main diagram, there are several Not Laws worth considering. The last rule produces Not Laws for O on laps 3 and 8. If J cannot swim laps 4 or 9, then O cannot swim laps 3 or 8.

The third rule, M O, creates two Not Laws because the rule applies to the first lap O swims.

Hence, O cannot swim lap 1 (and lap 6) and M cannot swim lap 5 (and lap 10). And, if O cannot swim laps 1 or 6, by applying the last rule we can deduce that J cannot swim laps 2 or 7.

The only remaining Not Law is L on laps 5 and 10. If L swims laps 5 and 10, the OJ block must swim laps 2-3 and 7-8. M must then swim laps 1 and 6. The only remaining laps for K to swim are laps 4 and 9, and thus K would swim immediately before L, a violation of the first rule.

A close examination of the game reveals that the placement options of the OJ block are limited. In fact, O and J can only be placed into three separate positions: 2-3 and 7-8; 4-5 and 9-10; 5-6 and 1-10. One approach to the game would be to Identify the Templates:

J02_Game_#4_setup_diagram 3.png

J02_Game_#4_setup_diagram 4.png

Although the above templates make the game quite easy, the original setup to the game can also be used to attack the game effectively.

[srcline@noctrl.edu]

Hello

I am completely lost on how to even approach this game.I went back and set up two rows because the swimmers repeat the laps. So I had

1 2 3 4 5

6 7 8 9 10

The only rules that I understood was the second rule, in that If jacobson does not swim laps 9 he does not swim lap 4.
With the first rule I diagrammed it as L>K and got that K cant swim laps 1 or 6 and L cant swim laps 5 or 10
With the third rule I diagrammed it as M> O1st lap
With the fourth rule I combined the third rules and got M>O_J

Please help with this game.
Thankyou
Sarah

[Clay Cooper]

Hi Sarah,

This game is much simpler than it looks, but don't be discouraged - it does seem a bit foreign at first.

Let's first observe that if the order of the swimmers in the first five laps is identical to the order of the swimmers in the second five laps (as we are told is the case), then a ten-blank game simplifies almost completely to a five blank game; we can almost (we will deal with the almost momentarily) disregard the second set of five laps and just treat the game as follows:

_ _ _ _ _ swimmer
1 2 3 4 5

That diagram will almost completely suffice for (and simplify the interpretation of) all of the rules except the last rule. For instance, the first rule will simplify to K cannot be immediately before L, or a KL not block, with the one additional provision that K cannot be last if L is first (since this would result in K racing 5th and L 6th, which would also break the rule). Does that make sense?

The last rule is a bit more complicated because we are told that at least one of J's laps is immediately preceded by a lap by O; this rule cannot be completely accounted for in a five-blank setup unless we treat it as a special case and sort of branch that set of possibilities off to the side. In other words, this rule is simply indicated by an OJ block (which results in both of J's laps being immediately preceded by a lap by O, which satisfies the rule) AND by an additional setup in which O is last and J first, since this would be the only other case in which this rule is satisfied, as it would result in O racing 5th and J 6th (paralleling the special case involving K and L mentioned above).

Hopefully this explanation has clarified the game for you somewhat; don't be overwhelmed by it, it is simpler than it looks. Good luck!