In the first section of this guide about solving specific game types, we described the strategy for solving problems requiring the arrangement of the game’s elements in linear fashion, either from up to down or, more typically, from left to right. 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, that is, not in a straight line.

What kind of spatial arrangements might you be asked to determine? 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.

What type of strategy is used to work through these kinds of games? Like the strategy for linear sequencing games, you will need to create a diagram on which you will place the available information. 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. Next, we will look at map games, showing how to create a diagram that allows the straightforward solving of these types of games, again concluding with a summary of strategy.

### Circular Games

Again, 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, this could 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. This helps in interpreting the relationships between elements, especially when you are given conditions that describe elements as being directly across from one another. It becomes much easier to see the spatial relationships if you use spokes.

Let’s look at a sample problem and try to set up a useful diagram.

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## Sample Game One

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. Now, what type of diagram is required? Obviously, we need a circle to represent the table. Additionally, we are told that there are eight chairs, evenly spaced. (Although we must keep in mind when solving the problem that only seven of these chairs will be occupied.) We will represent these as spokes coming off the central circle. Your initial diagram should look like this.

The next step is to summarize the given conditions, which we can write to the side of our diagram. 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.

Now, unlike linear arrangements, in a circular arrangement 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. Because of this ambiguity, it is especially critical to approach these problems systematically.

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. Do we know anything about exactly which seat either of them or in? No, we only know their relative positions. (Again, this is the critical difference between non-linear spatial games and their linear counterparts. The non-linear games rely much more on relative placements, rather than positioning an element into precise, certain locations.) 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.

Question One

(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?

Well, according to our diagrams, 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, that is, whether Ben is to the right of Antonia or vice versa, we could place the two of them into either of the two open positions, creating two arrangements for each of our diagrams, for a total of four arrangements. Rather than creating two additional schemes, however, it may be more useful to simply add this information as A/B and B/A for the two seats. These terms represent the possibility that either A or B is sitting in one position and either B or A is in the second position.

Now that we have this information placed onto our diagram, let’s work through each answer choice one at a time, remembering that we are looking for the condition that must be true. Choice A places Antonia between Ben and Denise. Is this possible? Yes. Since we are looking for the condition that must be true, however, we must also see if we could arrange the elements in an alternate way. Here are two 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. Is this possible? Yes. Can we come up with an arrangement in which this condition is not met? Again, yes. 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. You can see this by looking at the above two diagrams. (It remains true if you place A and B together onto the original alternative diagram we created, as well.) Therefore, choice E is the correct answer because this arrangement must always be true.

Question Two

(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. In looking at our two diagrams, we can see immediately that this condition can only be placed onto our diagrams into certain of the positions, since we need pairs of seats that do not already have someone assigned to them. Again, 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 asked to determine which of the answer choices are not possible arrangements. Again, let’s work through each individual answer choice, determining whether each is possible within the confines of our diagrams. 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.

### Summary of strategies for solving circle games

• Create the roster of elements.
• Once you have determined that you are working on a circle game, draw a circle for the diagram and represent the individual positions with lines, or spokes off that circle.
• Summarize the rules and write them to the side of your diagram.
• In determining which elements to place onto the circle first, begin with the element for which you have the most information. If you can determine elements that are across from each other, start by placing those onto your circle. Elements that can be placed in multiple positions can be represented either by alternative diagrams or simply use the same diagram and represent the either/or possibility with a symbol, such as A/B to represent that either A or B could be in that position.
• Read the question. Summarize symbolically any new information given to you. Try to place that new information onto the diagram. Determine whether the question is asking for a specific fixed arrangement (such as “who is sitting across from Howard?”), or a possible position, such as whether or not Louis could be to the left of Sam. Be sure to read carefully, and know whether you are being asked to look for the arrangement that is possible, the arrangement that is required or necessary, or the arrangement that is not possible.

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