Non-linear spatial arrangement
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
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.
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.
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.
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
Eduardo is sitting directly across
Gavin is sitting directly across
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
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
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
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 this 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? 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
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
Let's try another question for
this same game.
(2) If Antonia is sitting across
from the empty chair, which of the following cannot be true?
(A) Antonia is sitting next to
(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
· Create the roster of
· 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
1: Ordering Games
2: Characteristic Games
3: Grouping Games
4: Network Games
5: Non Linear Spatial Games
6: Map Games
GAME TYPE 6: Map Games