The Tenth Circle of LSAT Hell: Tiered Circular Games

    LSAT Logic Games | LSAT Prep

    Screen_Shot_2014-08-04_at_8.48.39_AMFollowing up on my earlier blog post about esoteric Logic Games that are having a comeback (just like Lindsay Lohan, if she were an esoteric logic game), today our focus is Circular Linear games.

    A game of this type appeared on the February 2014 LSAT, and it is rumoured to have been a "tiered circular linear" game, presumably involving two variable sets to be distributed around a circle. We'll never know for sure, but such a game is not outside the realm of possibility. What are the chances of getting a game like this? Slim to none, or so everyone thought. After all, the following lists every single appearance of a Circular Logic game on the test since 1991:

    • June 1991, Game #1
    • June 1993, Game #3 (OK, that was more of a square than a circle, but... same idea)
    • February 1999, Game #3
    • September 2003, Game #4

    Clearly, Circular Linear games are exceptionally rare, with a frequency rate of less than 1.5%. But... better safe than sorry. To demonstrate what the February test-takers may have had to deal with, I came up with a sample scenario of a Tiered Circular Linear game. Bear in mind: we have not seen an actual copy of February 2014 LSAT, and probably never will. Such is life.

    Three adults—F, G, and H—and three children—P, R, and S—are seated, evenly spaced, around a circular table. Any two of them are said to be sitting directly across from one another if and only if there are exactly two other people sitting between them, counting in either direction around the table. The following conditions govern the seating arrangement:
    Each child sits directly across from one of the adults.
    F does not sit immediately next to G.
    P does not sit immediately next to R.
    H does not sit immediately next to S.

    Like all circular games released thus far, this one consists of a fixed number of variables assigned to spaces evenly distributed around a circle, or a table. Notice that for each seat at the table we need to keep track of two attributes: age (adult vs. child) and name (F, G, H, P, R, and S), hence the name "tiered" circular game. Thankfully, the two variable sets are fixed to each other, and so we do not need to determine who is an adult and who is a child. Using tiers may not be necessary.

    To start with, do not waste your time drawing out a table. Use a “spokes” diagram instead, with each seat represented by the end of a “spoke.” Since there is an even number of individuals seated around the table, everyone must sit directly across from someone else. A “spokes” diagram will help you represent this idea more easily:

    Screen_Shot_2014-08-04_at_7.47.39_AM

    In Circular games with an even number of variables, rules involving opposites are particularly important, which is why the first rule (“Each child sits directly across from one of the adults”) is worth a closer look. Since there are as many children as there are adults, we can infer that - conversely - each adult sits directly across from one of the children. And, because the seats are not assigned specific numbers, we can place the adults and the children across from each other in an arbitrary fashion, as long as the arrangement does not violate the rule in question.

    By focusing on the categorial variable set (adults/children), you will quickly realize that there are only two possible ways to distribute them around the table:

     

    Screen_Shot_2014-08-04_at_7.56.24_AM

     

    This approach has the added benefit of restricting the variables that can occupy any given seat to one of three, rather than one in six. Since two templates are incredibly restrictive, we can examine more closely the application of the last three rules in the game:

    F does not sit immediately next to G.

    The second rule establishes that F does not sit immediately next to G. Since F and G are both adults, this rule would never be violated in Template 2. In Template 1, however, we need to ensure that F and G are never seated next to each other by placing another adult (H) between them:

    Screen_Shot_2014-08-04_at_8.01.55_AM

    On to the next rule:

    P does not sit immediately next to R.

    The third rule establishes that two of the children - P and R - do not sit next to each other. As with the previous rule, this rule would never be violated in Template 2, due to the alternating arrangement in that template. However, we need to ensure that P and R are not immediately adjacent in Template 1, and so a third child - S - must separate them, as shown below:

    Screen_Shot_2014-08-04_at_8.34.08_AM

    Last rule:

    H does not sit immediately next to S.

    The last rule forbids S (a child) from sitting immediately next to H (an adult). Clearly this would never be a problem in Template 1, because H and S sit directly across from one another in that template (see above). However, we must ensure that the rule is not violated in Template 2, where the adults and children alternate seats. If S cannot sit next to H, the two children sitting next to H must be none other than P and R; conversely, the two adults sitting next to S must be F and G:

    Screen_Shot_2014-08-04_at_8.38.43_AM

    These two templates provide a powerful insight into the relative placement of the six variables. They also reveal a critical inference, namely, that H and S must always sit directly across from each other.

    Hopefully, this post convinced you that Circular games—even Tiered Circular ones—are hardly the Tenth circle of hell they are rumored to be. For those enrolled in our Full Length or Live Online LSAT courses, Circular Games are discussed in the Lesson and Homework Supplements to Lesson 3. We also discuss Circular Games in the Fogotten Few chapter of our famous Logic Games Bible.

    Good luck!