Map games
A map games requires the arrangement of the game's elements on a plane according to their relative compass directions (north, south, west, and east). These games, like the circular games we just looked at, are similar to the linear spatial arrangement games we've already studied. Like those, we are ordering elements according to their relative positions. The complication is that there are two criteria for ordering, a north-south component and a west-east component. These two ordering processes must be considered simultaneously, which is why these problems are challenging.
Let's begin by working through a sample game.
Game Two
Within a county, there are six cities, L, M, N, O, P, and Q. The location of these cities relative to each another is as follows:
Q is further north than P.
P is further west than O.
L is further north than P.
M is further east than P.
N is further west than O.
As with our previous game, we can begin with the roster, and them summarize the rules. (Roster = L, M, N, O, P, and Q.) Next, we are going to try to construct a diagram to enable us to solve this problem. First, however, we must be sure to read the premise and conditions extremely carefully and precisely. For example, when the problem states that Q is further north than P, what exactly does that mean? Obviously, Q is more north than P, but does it mean that Q is directly north of P? No. Q may be further west or east than P, and we cannot determine that from the information given to us. Like the circular arrangement games, map games often involve conditions that describe relative spatial arrangements, not fixed positions. You could approach this as a linear sequence game in which you are aligning elements in two linear sequences, north to south and west to east, and then attempt to merge these two sequences. That approach however, is quite complicated, and could easily result in the construction of an inaccurate map, which will not help us answer the questions. Instead, we will try to construct a map that indicates the relative positions of the elements.
The first step in constructing such a map is to determine the most frequently mentioned element. In this case, P is mentioned four times in the conditions, so we can start with this element. (If instead let's say that P and O were each mentioned the same number of times, we could simply pick one to begin.) Once we decide which element is mentioned most, we will construct a simple quadrant map, and place that element at its center. For simple quadrant maps, the y-axis represents the north-south direction, and the x-axis represents the west-east direction. (On this quadrant map, north will be at the top, south at the bottom, west to the left, and east to the right.)
Next, try to place on our new map all of the conditions that describe elements and their relationships to the central element. In this case, our central element is P, and the conditions that describe elements and their relationship to P are: Q is further north than P, P is further west than O, L is further north than P, and M is further east than P. Let's put each of these conditions onto our map.
When you place these elements, you should have realized that you do not know the positions of the other elements relative to each other. That is, which is further north, Q or L? Is Q west or east of L? Likewise for O and M. This is a critical point: Remember that at this stage, you are placing elements on the map with regards to the central element only. It is for that reason that it is useful to place these elements close to the north-south or east-west axes, to remind you that their exact location is not known. Another strategy would be to mark the elements whose exact locations are unknown, by dotted lines, shading, or parentheses.
After having placed the central element, and positioning the other elements specifically related to that element in the conditions, the next step is to look at the other conditions and determine how to translate that information to the new map. In this case, we have the statement that N is further west than O. Where is N with regards to P? Is it west or east? We don't know. Because we only know its relationship to O, we will place it on our map west of O, arbitrarily place it either west or east of P, but be sure to draw an arrow from O to N indicating that it is this relationship which has caused us to place N in this position. (As opposed to a relationship with the central element, P.)
Now that we have constructed this map, let's look at the questions.
Question One
(1) A person leaving N and driving in a southwest direction on a single, straight road, might pass by which of the following cities, in the order given?
(A) M, L, O
(B) L, P, M
(C) L, M, Q
(D) O, M, P
(E) L, P, Q
Now, if we try to answer this question by looking at the map, it can seem quite confusing. If our map indicated exact, precise locations, it would appear that the only other city that could be encountered on a southwest route from city N would be city P. This is not one of the choices. What exactly do we know about the relative positioning of N? Well, we know that it is west of O. Therefore, a road leading from N in the southwest direction could never encounter O. Thus, we can eliminate all answer choices that include O, choices A and D. Next, let's tackle the remaining answer choices, one at a time, using what we know about the relationships between P and the other cities. Choice B offers L, P, and M. What do we know about L compared to P? L is north of P, so encountering L before P would make sense if a person were to travel in a southwestern direction. What about P compared to M? If we look at our map, we see that M is east of P, not west. Therefore, this linear sequence of cities would be impossible, and we can eliminate choice B. Next let's look at choice C: Q, M, then L. This one does not contain any of our fixed elements, that is, it does not contain our central element, nor does it contain the two other two elements for which we have relative location information. Because of this, we are unable to rule out this option. (Indeed, this is actually the correct choice.) Let's look at the last choice, E, which has the sequence L, P, and Q. Again, let's start by looking at the relationship between L and P. We know that L is north of P, which is consistent with L being encountered first. What about P and Q? We know that Q is actually north of P, which is not consistent with it being encountered after P on a southwestern route. Therefore, we can eliminate this possibility. To confirm that choice C is correct, let's see if we can rearrange the map to reflect the necessary route (southwest from N) while still maintaining the appropriate relationships between the central element, P, and the surrounding cities.
Indeed, we can maintain the necessary relationships to P (and that of N to O, represented still by the arrow) and allow a southwestern route from N to contact these three cities in the order described in choice C.
Let's try another question.
Question Two
(2) If city Q is located directly north of city N, then which of the following statements must be true:
(A) City P is further south than
city O.
(B) City Q is further west than city P.
(C) City O is further east than city Q.
(D) City N is further north than city P.
(E) City L is further south than city O.
To begin addressing this question, first add the new information to the map, that is, we must either reposition city N, or reposition city Q. When we do this, we must also draw an arrow indicating their known relative positions (that city Q is directly north of city N).
Does our new map indicate that Q is east of L? Or that Q is north of O? No. The only spatial relationships that we know are the relationships of the cities (excluding N) to P, and the relationships between N and O, and Q respectively. Before we work through the answer choices, keep in mind that the correct answer will reflect either a relationship described in the original conditions or it will reflect the new condition given in the question. If we look through the answer choices according to the original conditions only, we cannot eliminate any of the choices, all relationships are possible. Therefore, the correct answer must reflect the new relationship. (You should go through each of the answer choices to confirm this for yourself. Remember, if you can draw a map showing the relationship described that does not contradict the original conditions set out in the initial premise, then the spatial relationship you have constructed cannot be ruled out.) What spatial relationships are affected by knowing that Q is directly north of N? Do we know anything about the relationship of Q to any other cities? Only that Q is north of P, so Q is north of both N and P. From that, can we deduce anything about the relationship of N and P? No, so we can eliminate choice D for this reason. We do know that N is west of O. If Q is north of N, it too must be west of O (and O is east of Q). This is the correct choice, C, the only answer choice that describes a relationship that we can confirm as true given the information provided.
Summary of strategy for map games