Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Circular Linearity, Identify the Possibilities game.

The first rule establishes that at least one of any three consecutively numbered lights is off, meaning three lights in a row cannot be on:

PT8-Jun1993 LG Explanations game 2 setup diagram 1.png

The second rule establishes that light 8 is on:

PT8-Jun1993 LG Explanations game 2 setup diagram 2.png

The third rule states that lights 2 and 7 cannot be on when light 1 is on:

1 2, 7

This rule will play a pivotal role in an inference to be discussed shortly.

The fourth rule indicates that at least one of the three lights on each side is on:

PT8-Jun1993 LG Explanations game 2 setup diagram 3.png

The fifth rule is another rule about sides and lights, and it indicates that if exactly one light on a side is on, then that light must be the center light:


Side 1 light on Center on

The contrapositive of this rule is:

Center on Side 1 light on

Since a side must have at least one light on and cannot have all three lights on, this contrapositive can be translated as:

Center off Side 2 lights on


When a side has two lights on but the center is not on, then both corners must be on:


Center off Both corners on that side are on


The contrapositive of this inference is:

Both corners on that side are on Center on


Thus, if one of the corners is off, then the center light is automatically on.

The final rule states that two lights on the north side are on. From the third rule we know that lights 1 and 2 cannot be on at the same time, so, by Hurdling the Uncertainty we can infer that light 3 must always be on (otherwise you could not fulfill the constraints of this rule):

PT8-Jun1993 LG Explanations game 2 setup diagram 4.png

At this point, most students move on to the questions. But, there are six rules, and several of those rules establish general limitations on each side or section of three lights, and these rules, when combined with the fact that the status of two of the eight lights is already determined, indicate that the game cannot have a large number of solutions. The best decision, then, is to explore Identifying the Possibilities.

Start first with the third rule, which states that lights 2 and 7 are off when light 1 is on. By turning light 1 on, lights 2 and 7 automatically are off, leaving lights 4, 5, and 6 undetermined. But, from our discussion of the fifth rule, when a corner light is off (as light 7 is), then the center light on that side is on. Hence, light 6 must be on. Lights 4 and 5 cannot be precisely determined, but if one is on, the other is off (if both were on, the first rule would be violated), leading to a dual-option. Combining all of the information gives us only two possibilities when light 1 is on:

Template #1:

PT8-Jun1993 LG Explanations game 2 setup diagram 5.png

Of course, light 1 could be off. In that case, light 2 must be on in order to meet the constraints of the final rule. With lights 2 and 3 on, light 4 must be off in order to conform to the first rule. With light 4 off, light 5 must be on in order to abide by the fifth rule. The only undetermined lights are 6 and 7, but both cannot be on (otherwise the first rule would be violated) and both cannot be off (otherwise the fifth rule would be violated). Thus, one of lights 6 and 7 is on, and the other is off, leading to two possibilities:

Template #2:

PT8-Jun1993 LG Explanations game 2 setup diagram 6.png

Thus, because all possibilities have been explored when light 1 is on and when it is off, and light 1 has no more possible positions, we have explored all possibilities of the game, and there are only four possible solutions, as captured by the two templates above.

[hannah.zale]

My question is regarding June 1993 Game #4 (stop lights, on/off), which is listed in the Logic Games Problem Set #1.

I don’t think this is supposed to be a hard game, but I am unfamiliar with the type, and not sure how to do the set-up quickly/effectively. The game comes with a diagram of the street lights, placed around a city block. Advise on how to best diagram the rules and inferences and attack the game efficiently would be greatly appreciated.

Thanks so much!

[Dave Killoran]

Hi Hannah,

Thanks for the question. I think you actually mean Game #2 from that test, which features 8 lights around a square parking lot, and each light is either on or off.

Basically, this is a an unusual game that has linear elements controlling the action. In fact, the rules are so restrictive that there are only a limited number of solutions to game, and two major templates that control the action. The rule about at least one of three consecutive lights being off, and the second to last rule interact in a very powerful way.

The way to diagram this game is to use their diagram as the template, and just put O for on and not-O (O with a slash through it) for off. Thus, for the second rule, just put an O next to light 8.

There is a key inference in the game. The final rule states that two lights on the north side are on. From the third rule we know that lights 1 and 2 cannot be on at the same time, so, by Hurdling the Uncertainty we can infer that light 3 must always be on (otherwise you could not fulfill the constraints of this rule). This inference answers question #7 and helps to answer several others.

Without analyzing each rule, let's discuss the restrictions in the game. There are six rules, and several of those rules establish general limitations on each side or section of three lights, and these rules, when combined with the fact that the status of two of the eight lights is already determined, indicate that the game cannot have a large number of solutions. The best decision, then, is to explore Identifying the Possibilities.

Template #1

Start first with the third rule, which states that lights 2 and 7 are off when light 1 is on. By turning light 1 on, lights 2 and 7 automatically are off, leaving lights 4, 5, and 6 undetermined. But, when a corner light is off (as light 7 is), then the center light on that side is on. Hence, light 6 must be on. Lights 4 and 5 cannot be precisely determined, but if one is on, the other is off (if both were on, the first rule would be violated), leading to a dual-option.

Template #2

Of course, light 1 could be off. In that case, light 2 must be on in order to meet the constraints of the final rule. With lights 2 and 3 on, light 4 must be off in order to conform to the first rule. With light 4 off, light 5 must be on in order to abide by the fifth rule. The only undetermined lights are 6 and 7, but both cannot be on (otherwise the first rule would be violated) and both cannot be off (otherwise the fifth rule would be violated). Thus, one of lights 6 and 7 is on, and the other is off, leading to two possibilities.

Thus, amazingly, there are only four solutions in this game (two solutions in each template).

That's a start, but please let me know if you have any other questions.

Thanks!

[smile22]

The explanation above is very helpful. I do have a follow up question to your explanation, however. In your above explanation you state that if a corner light is off, then the center light on that given side must be on. I am a bit confused as to why the center light would have to be on based on the rules given in the problem. Could you please explain?

Thanks!

[connorjstone]

I tried for so long to set up this game but could not figure it out.
This game is found under the Logic Games Problem Set 1 under the Supplemental Problem Sets.
I am most confused by the first rule which states, "At least one of any three consecutively numbered lights is off."
If I could get some help with this setup, that would be awesome.
Thanks

[marcnash]

Hi,

do we expect to see such game in future LSATs? I think it's kinda different than what I have been practising recently so I am wondering whether I should spend some time on it or not

[Nikki Siclunov]

Hi Marc,

Whether a game like this is likely to be given on a future LSAT is anyone's guess, and predicting what the LSAC will do is an exercise in futility. That said, in the last few years there has been an unusually high occurrence of long-forgotten game types (e.g. Pattern) or else of games that just seem crazy or unusual (similar to the one you asked about). Do I expect to see a similar game on a future test? No. What I do expect to see is games that attempt to surprise and confuse you. Rest assured, however, that if you know the fundamental principles of approach - you know how to identify limited solution sets, for instance, or how look for points of restriction, make hypotheticals and/or local diagrams, etc. - you shouldn't have much trouble with such games. But, you should definitely expect... the unexpected.

Sorry if this is not what you wanted to hear Let me know if you have any questions.

Thanks,

[jlam061695]

Can someone please explain how the fifth rule of this game functions? I do not understand why the center light must be on if a corner light is off? It is not stated anywhere in the rules, and I do not know how that inference is derived based on what is given.

[Dave Killoran]

Hi Jlam,

Thanks for the question! The fifth rule is a difficult one, and as you might expect it has a big impact on the game. So, let's take a closer look at this one.

The fifth rule states, "If any side has exactly one of its three lights on, then that light is its center light." The first word is "if," and thus this is a conditional rule. The initial diagram appears as:


Side 1 light on Center on


The contrapositive of this rule is:


Center on Side 1 light on


In isolation, the rule and its contrapositive look relatively straightforward and the tendency here is to simply make the diagram and then move onward. But, whenever you have conditional rules in play and variable sets with just two or three options, always consider each condition when you take the contrapositive and attempt to see if the meaning of one or both isn't impacted by the limited number of options. Side note: that consideration doesn't have to be written out (and usually isn't unless there is a viable inference); just make a quick mental calculation and if nothing is there, proceed, and if there is something then explore it further.

In this case, there is something worth looking at more deeply. Each light has only two options: on and off. So, when the contrapositive says that the center light is not on, that is identical to saying it is off:


Center on = Center off


That's fairly easy to understand, but for most people it's nice to take the negative out of the condition so the translation is useful just on that front. We can perform a similar translation on the necessary condition. The necessary condition in the contrapositive indicates that a side does not have just one light on. So, because there are three lights on a side and each side must have at least one light on (from the fourth rule), that means that the side must have two or three lights on. However, the first rule effectively states that you can't have all three lights on on a side, and so this condition translates to exactly two lights on a side must be on:


Side 1 light on = Side 2 lights on


Ok, that looks good so far, but wait, there's more! If there are exactly two lights on a side that are on, and we know that the center light is off, then we can infer that the two lights that are on are the corner lights, leading to this translation:


Side 1 light on = Side 2 lights on = Both corners on that side are on


Adding the above together, this allows us to show an equivalence diagram for the contrapositive above:


Center off Both corners on that side are on


If you reached this point of analysis during the game, congrats! It shows that you recognized the limitations in the variable sets, and used those restrictions to reveal deeper truths about the game. However, we can take one more step here, and it's quite a useful one. When we look at the contrapositive of the fifth rule, and then how we re-formulated that rule by looking at the impact each condition had on the related variable sets, the contrapositive and its translation look rather different. They have the same meaning, but they express different aspects of the relationship. Because of this, consider the contrapositive of our translated contrapositive. Our translated contrapositive was:


Center off Both corners on that side are on


To take the contrapositive of this, we reverse and negate the terms as usual. The first condition (Center off) is easy and becomes Center on. The other condition, "Both corners on that side are on" becomes "Both corners on that side are on" which really means "at least one corner is off," and thus the contrapositive is:


at least one corner is off Center on


So, operationally, there are two big takeaways from this rule:

• 1. If the center light is off, then both corners are on.
  
  2. If one of the corners is off, then the center light is on.

Overall, it's a really tricky rule, but a great example of how limitations in the number of options (lights = on/off, and only three lights per side) can create a chain of enlightening inferences. Study the above translations because it is the kind of thing you see on tough LSAT games but the process itself is not that tough once you start doing it. The hard part is really just to remember or know that you should begin applying this process of analysis. The tipoff? Limitation in the number of options.

Please let me know if that helps. Thanks!

[jlam061695]

Dave,

Thanks for answering my question! Explaining the rule in terms of contrapositives helps a lot!