Note: Circular Games are rare on the LSAT (there hasn’t been one since 2003). You’re unlikely to get one on test day. So, consider skipping ahead or circling back to this lesson only if you have free time.
In this section, we will describe how to solve games that require you to define the spatial relationships between elements that are arranged in a non-linear fashion (not in a straight line).
Elements may be arranged circularly, like the arrangement of people sitting at a round table or you might be asked to arrange elements according to their positions on a map, by the compass directions, north, south, west, and east.
The best diagram will be one that accurately represents the premise of the game, whether it is the circular arrangement of elements or the arrangement of points on a plane, such as a map. In this section, we will begin by looking at a game requiring the circular arrangement of elements. Through the creation of an effective diagram, we will work through sample questions and conclude with an overall strategy for working through these games.
The best diagram for each game is one that represents the game’s premise. For example, if you are asked to determine the seating arrangement around a table, you will use a circle as the basis of your diagram.
Since we are arranging seats around the table, it can be represented in multiple ways such as single points on the circle or lines representing the individual chairs. A better way is to draw spokes or lines off the circle.
Let’s look at a sample problem.
Next LSAT: September 16
Seven friends- Antonia, Ben, Carlos, Denise, Eduardo, Felicity, and Gavin- go to a restaurant for dinner. They are seated at a round table with eight chairs, evenly spaced. Each chair is directly across the table from exactly one other chair, one of which remains empty. Felicity is sitting between Denise and Carlos. Eduardo is sitting directly across from Denise. Gavin is sitting directly across from Felicity.
First, we must create a roster of the game elements. In this case, the roster is the list of seven friends, which we can represent as A, B, C, D, E, F, and G.
Additionally, we are told that there are eight chairs, evenly spaced. We will represent these as spokes coming off the central circle. Your initial diagram should look like this.
The first statement, Felicity is between Denise and Carlos, can be written as DFC or CFD since we do not know the exact placement of Denise and Carlos with respect to Felicity (who is on her left and who is on her right), we only know that one is on either side.
The second statement, Eduardo is directly across from Denise, can be written as E<–>D, where the arrow indicates that the two elements, E and D, are across from each other. The final condition, that Gavin is across from Felicity, can similarly be represented as G<–>F.
There is no definite starting or ending place. In this example of eight chairs around a table, a chair that is one place away from another chair counting in one direction is also seven places away from that same chair if you are counting in the opposite direction.
Where should we start in placing the information from the conditions onto our diagram?
A good guiding rule is to look for the element whose placement we know the most about. In this case, we know two pieces of information about Felicity’s position at the table, so we can use her as our starting point. Let’s start with the last condition first, that she is seated across from Gavin.
Therefore, we can place Felicity and Gavin into any two of the positions in our diagram, as long as they are positioned directly across from each other.
We also know that Denise and Carlos are on either side of Felicity, though we do not know who is seated to her right and who is seated to her left. Therefore, we should create two diagrams, reflecting these two possible arrangements.
Now that we have these two possibilities, let’s look at the remaining condition, E<–>D. Since we have D positioned on both of our diagrams, we can position E accordingly.
Now let’s tackle some sample questions for the game.
(1) If Ben is sitting next to Antonia, which of the following must be true?
(A) Ben and Denise sit on either side of Antonia.
(B) Ben is sitting directly across from Carlos.
(C) Felicity and Gavin are on either side of Carlos.
(D) Antonia and Denise are on either side of Ben.
(E) Eduardo and Carlos sit on either side of the empty chair.
We are given an additional condition in the question: Ben and Antonia are sitting next to each other. We can summarize this as BA or AB. Let’s add this to our list of conditions (for this problem only). In looking at our two diagrams, what can we deduce immediately about where Ben and Antonia must be sitting?
According to the diagram, there are only two open seats that would allow Ben and Antonia to be sitting next to each other. However, since we cannot know who is on which side, it may be more useful to add this information as A/B and B/A for the two seats – it represents these multiple possibilities.
Choice (A) places Antonia between Ben and Denise. Is this possible? Yes. Since we are looking for the condition that must be true, we must also see if we could arrange the elements in an alternate way.
Two other possibilities: the left diagram allowing the positioning described in choice (A) and the right diagram showing an arrangement in which this condition is not met.
Therefore, we can eliminate choice (A).
In choice (B), Ben is across from Carlos. We can see these two possibilities by looking at the previous diagrams, so we can eliminate choice (B). In choice (C), Carlos is between Felicity and Gavin. Is this possible? In looking at our diagram, we can see that this cannot work. Felicity and Gavin are directly across from each other; there are three seats in between them. In choice (D), Ben is between Antonia and Denise. Again, by looking at the previous set of possible diagrams, we can see that this is a possible arrangement, though not the only arrangement we could construct. Therefore, we can eliminate choice (D). In choice (E), we are told that the empty chair is between Eduardo and Carlos. Is this possible? Can we place all the elements onto the diagram and have the empty chair between Eduardo and Carlos? Yes. Is it necessary to have this placement? In this case, the answer is yes. Thus, choice (E) is the correct answer because this arrangement must always be true.
(2) If Antonia is sitting across from the empty chair, which of the following cannot be true?
(A) Antonia is sitting next to Carlos.
(B) Eduardo is across from Denise.
(C) Carlos is sitting between Felicity and Antonia.
(D) Gavin is sitting next to Antonia.
(E) Gavin is sitting next to Eduardo.
For this question, we are given the additional condition that Antonia is sitting across from the empty chair. We can represent this as A<–>emp. We can represent the possibilities as A/emp and emp/A to represent the idea that either element can be in either place on our diagram.
We are tasked to determine which of the answer choices are not possible arrangements.
For choice (A), can Antonia be next to Carlos? Yes, we can see that this is possible from either of the two diagrams. For choice (B), can Eduardo be across from Denise? Yes, this is actually one of the original, required conditions. For choice (C), can Carlos be between Felicity and Antonia? Yes, this condition can be met with either of the two diagrams. For choice (D), can Gavin be next to Antonia? According to our diagram, this is not a possibility. If we were to place Antonia next to Gavin, either to his left or his right, she would then be across from Carlos instead of being across from the empty chair. Therefore, this arrangement is not possible.
To be sure, let’s check choice (E), which states that Gavin is sitting next to Eduardo. Since we know that Eduardo is across from Denise, we have already placed him onto the diagram into the seat to Gavin’s left or right. Therefore, this condition is possible (and mandatory). Choice (D) is the only one that cannot be true.
Next LSAT: September 16