VI. GAME TYPE 3: GROUPING GAMES
Grouping
games are another common type of game found on the LSAT . Unlike
ordering games, which require you to organize the elements or
subjects of the game into either a spatial or sequential order,
grouping games require you to divide the elements of a game into
two or more groups. Often, you will be asked to separate the
elements into groups based on a characteristic, and it is for
this reason that these games are similar to the characteristic
games that we described earlier. However, you will see as we
work through the sample games in this section that these games
are distinct from characteristic games in that rather than assigning
or matching characteristics to single elements, we are instead
grouping elements that share characteristics. The distinction
between these two types of games in what you are being asked
to do, as well as the different strategy required for solving
these problems, will become apparent once we start working through
a few of these games.
We
will begin this section with a typical grouping game like those
that you might encounter on the exam. We will then show how to
tackle this problem by applying the basic game solving strategy
that we outlined in the first part of this chapter as we create
a diagram on which we will summarize the given information from
the game. Finally, we will then work through an additional sample
game by applying the strategy to successfully answer the questions.
Sample Grouping Game One
Ten freshmen students, L, M,
N, O, P, Q, R, S, T, and U, arrive at Egghead University for
their first term of college. They are told that they have been
assigned to one of three dormitories: Genius Hall, Intellectual
Center, or Brainy Building. At least three of the ten students
are assigned to each dormitory.
M, P, and U are all assigned
to separate dormitories.
L, N, and T are all assigned to the same dormitory.
Both O and R are assigned to Intellectual Center.
First, what are we being asked
to do here? Out of a group of ten students, we need to separate
them into three groups based on the dormitory in which they will
live. Once we have read through the premise of the game, realized
it is a grouping game, and identified the task we are asked to
solve, the next step in our game strategy is to create the roster.
In this case, the roster of elements is the list of students.
Roster: L, M, N, O, P, Q, R,
S, T, and U
For grouping games, you may want
to create a second list of the groups into which you will be
dividing the elements. In this case, the groups are the three
buildings: Genius Hall, Intellectual Center, or Brainy Building.
Groups: Genius, Intellectual,
Brainy (g, i, and b)
Notice that we represent the
group names with lower case letters, rather than the capital
letters we used to represent the students. It's always a good
idea to make your abbreviations for the roster of elements and
the abbreviations for the groups look different so you don't
run the risk of mixing them up during the process of solving
the game.
Okay, according to our game-solving
strategy, our next step is to symbolize the conditions. In this
case, one of the conditions is actually contained within the
premise of the game: we are told that at least three students
are assigned to each of the three dormitories. What additional
information can we deduce from that statement? Well, if there
are ten students, and at least three will be in each dorm, that
would give us three students in three dorms, for a total of nine
students. Since we have ten students, not nine, we can conclude
that one of the dormitories will have four students, and the
other two will have three students each. We don't know yet which
of the three dormitories has four students, and which two have
three, but for now we can summarize this simply as 3-3-4.
The next condition (the first
in the list of conditions following the premise) tells us that
M, P, and U are all assigned to separate dormitories. This means
that if M is in a dorm, then neither P nor U can be in that same
dorm. (This applies to the others as well, that is, if P is in
a dorm, then M and U are not in that dorm, etc.) We can use the
same symbol we used earlier to indicate "not", the
^ symbol, and represent this condition as:
If M ^P, if P ^M
If M ^U, if U ^M
If P ^U, if U ^P.
As you can see, that one sentence
condition has given us a lot of information. We know that these
three students are assigned to the three different dormitories.
How else could we represent this condition symbolically? Rather
than writing three lines, a shorter way might be to just use
the ? symbol. In this case we can use it to represent the fact
that if one of the students is assigned to a dorm, the others
cannot also be assigned to that dorm:
M ? U ? P
The next condition tells us that
students L, N, and T are all assigned to the same dormitory.
Using the opposite symbol from that which we used for the previous
condition, we can represent this as:
L = N = T
Again, what other information
can we deduce from this condition? Since we know that two of
the dormitories will have three students and one will have four,
we know that either only these three students will live in one
dormitory, or one additional student could also be assigned to
the same dormitory and that this dormitory would then be the
one that has four students. Now, let's think about the previous
condition. We know that students M, U, and P all live in separate
dorms (M ? U ? P). That means that one of them, though we don't
yet know which one, will be assigned to the same dorm as L, N,
and T. Therefore, we can modify our condition to represent this
new information:
4 = L, N, T and (M, U, or P)
The third condition tells us
that both O and R are assigned to Intellectual Center. We can
represent this condition as:
i = O and R
Now, let's create a diagram to
help us think about the problem. What kind of diagram will best
represent the arrangement of elements in the game? Unlike ordering
games or network games, we don't need a complicated diagram.
We only need to represent our three groups and we can just use
three boxes for this. (Note: in this diagram the groups are labeled
with their full names for clarity, but on the actual exam, you
will probably want to use the abbreviations in order to save
time.) Within each group box, we will put three horizontal lines
for the three elements we will place in those groups. (Remember
that one of the three dorms has four students. Since we don't
know yet which one, we can represent that underneath each of
the four group boxes with a horizontal line and a question mark.
) Like our other games, we will write the conditions for the
game to the side.
Now that we have our diagram,
let's look at the conditions we have written at the side and
see which ones can be transferred into the diagram. As before,
let's start with the conditions that are fixed. Starting with
the last condition, we see that O and R will be in group i (Intellectual),
so let us write O and R in two of the lines in the Intellectual
group.
If we look at the second to last
condition we have written to the side, 4 = L, N, T and (M, U,
or P), we remember that L, N, T and one other student are in
the dorm that has four students. Because we already have O and
R in the Intellectual dorm, there is not enough space left in
that dorm. Therefore, we know that the Intellectual dorm does
not have four people in it, so we can eliminate the ? line from
underneath this group. Further, if we look at our second condition
which states that M, U, and P must be in separate groups, we
know that one of these students must be the third one in the
Intellectual group. We can represent this possibility as M /
U/ P on that third line, symbolizing that one of these must be
in this group.
The next step is to look over
the conditions again to see if any new information can be deduced
when considered together with your diagram. We know that L, N,
T and either M, U, and P make up the dorm that has four of the
students, but we don't know yet if these four are in the Genius
or Brainy dorms. It seems as though this is all the information
we can fill in so far.
Next, let's proceed to some sample
questions.
Question One
If M and Q are assigned to the
same dorm, which of the following cannot be true?
(A) O and U are in the same dorm.
(B) R and P are in the same dorm.
(C) Q and S are in the same dorm.
(D) U and L are in the same dorm.
(E) M and T are in the same dorm.
After reading the question, the
first step is to determine what additional information is provided
by the question itself. In this case, we are told that M and
Q are assigned to the same dorm. If we look at our diagram, M
and Q could fit into either the Genius or the Brainy dorms. If
we look at the answer choices, we see that we are not going to
be asked which students are in specific dorms, only which students
are assigned to the same dorm. So, let's arbitrarily insert M
and Q into the Genius dorm to start. Once we have placed M into
the Genius dorm, then we can remove it from being a possibility
in the Intellectual dorm. We can also add the possibility of
U or P into the Brainy dorm as well.
Now that we have added the new
information provided by the question to our diagram, let's review
the conditions we have on the right of the diagram and see if
we can deduce anything further about how to place the students
in the groups. Again, let's look at the requirement that L, N,
and P are in the group of four students. Is there room in the
Genius dorm for four students? No. Therefore, we must place the
set of L, N, and T into the Brainy dorm, and also move into that
dorm the fourth space, which would be held by U or P.
With this additional information
on our diagram, let's now look at the answer choices. Remember
that we are asked to figure out which of the following choices
cannot be true. For choice A, can O and U be in the same dorm?
Looking at our diagram, we see that O is in Intellectual. Can
U be in this group? Yes, we have it listed as a possibility for
the third spot. For choice B, can R and P be in the same dorm?
Again, R is in the Intellectual group, and P is listed as a possibility.
For choice C, can Q and S be in the same dorm? Well, we've placed
Q into the Genius dorm. Can S fill that empty spot? If we look
through the conditions, we don't see any restrictions on where
S can be. So that too seems like a possibility. (And remember,
we are looking for the choice that is not possible.) For choice
D, can U and L be in the same dorm? U is listed as a possibility
for the Brainy dorm, which is where we have placed L, so this
too is a possible arrangement. For choice E, can M and T be in
the same dorm? We have placed M in the Genius, and T in Brainy.
This is not a possibility, so choice E is the correct answer.
(Note: we arbitrarily placed M and Q into the Genius dorm. In
this case, it doesn't matter because both the Genius dorm and
the Brainy dorm have the same set-up, that is, they can either
have three or four students. If you are not sure about this,
work through the problem again with M and Q in the Brainy dorm
instead.)
Let's try a second question for
the same game. What do we need to remember before utilizing our
same diagram to answer the next question? Any information that
was given by the previous question must not be carried over to
the next question. Only information from the original premise
and conditions can be applied to each question.
Question Two
Which of the following must be
true for the group of ten students?
(A) Exactly three students are
assigned to Genius Hall.
(B) Exactly three students are assigned to Intellectual Center.
(C) Exactly three students are assigned to Brainy Building.
(D) Exactly four students are assigned to Genius Hall.
(E) Exactly four students are assigned to Intellectual Center.
We know that two of the dormitories
will have three students and one will have four. Do we know anything
about which of the three dormitories could have four and which
must have only three? Let's look at our diagram again. (Remember,
we need to look at our original diagram, not the one that we
made for the last question that contained additional information.)
We are asked about how many students
can be assigned to each dorm. As we've drawn it, the group of
four students could be in either Genius Hall or the Brainy Building,
not in Intellectual Center. Choice B, that exactly three students
are assigned to Intellectual Center, is the correct choice. Choices
A, C, and D are all possible arrangements that satisfy our conditions,
but we are asked which choice must be correct, not what is merely
possible. Choice E is not correct, since we know that only three
students are assigned to Intellectual Center.
Let's try one more question for
this game.
Question Three
If U is assigned to the same
dorm as S, how many distinct groups of students could be assigned
to Genius Hall?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
These particular types of questions
are quite common in the games section, and can be time-consuming
because we are asked to figure out how many possibilities will
satisfy a given arrangement. Let's start with what we know. In
the question, we are told that U and S are in the same dorm.
Now, we know that they can't be in Intellectual Center, since
we don't have enough spaces available to fit two students in,
only space enough for one additional one besides O and R. Therefore,
U and S must be in either Genius Hall or in Brainy Building.
Since we don't know which, we need to draw out both possibilities.
We will label these possibilities A and B, for locating them
in Genius Hall or Brainy Building.
Let's look at our conditions
again. We know we have the block of four to fit into a group.
Can the group of four students be in the same dorm as U and S?
No. Let's then put the group of four into the other building.
Remember, we need to do this for both possible arrangements,
A and B. When we do this, we see that we have one blank remaining,
which will can fill in with the one remaining student, Q.
So, we are asked how many distinct
groups of students could be assigned to Genius Hall. In possibility
A, where S, Q, and U are assigned to this dorm, there is only
that one set. In possibility B, which we have assigned to L,
N, T and either M or P, there are two possibilities (L, N, T,
and M or L, N, T, and P). Therefore, we have a total of three
possible arrangements of students in Genius Hall, so the correct
choice is C.
Before we work through another
sample game, let's review the basic strategy for solving grouping
games.
Grouping Game Strategy
· Read through the premise
of the game and identify the task you are asked to solve.
· Create the roster of
elements you are asked to place into groups.
· You may also want to
create a second list of the groups into which you will be dividing
the elements.
· Symbolize the conditions.
· Create a diagram that
will best represent the multiple groups into which we will arrange
the elements of the game. Within each group, represent the number
of elements that can be possibly placed in that group.
· Like the strategy we
use for other games, write the conditions for the game to the
side of the diagram.
· Look at the conditions
that are written at the side and see which ones can be transferred
into the diagram. Always start with the conditions that are fixed.
· Once you have transferred
the information from the conditions to the diagram, look over
the conditions again to see if any new information can be deduced
when considered together with your diagram.
· Read the question.
· Determine what additional
information (if any) is provided by the question itself.
· Add that new information
provided by the question to the diagram, and again review the
conditions on the right of the diagram and see if you can make
any more deductions about elements that can be put into a group
on the diagram.
· Solve the question by
consulting your diagram. If necessary, check each possibility
against your diagram.
· When you proceed to
the next question, remember that any information that was given
by the previous question must not be carried over to the next
question.
Let's try another sample game
(a little more difficult) and try to work through it implementing
this strategy.
Sample Grouping Game Two
In the front window of Peter's
Pet Palace, there are three cages- cage one, cage two, and cage
three- that contain a total of seven cats. Each cat is a different
color: brown, black, white, gray, tabby, calico, or orange.
The gray cat and the tabby cat
are not in the same cage.
The black cat is in cage three only if the white cat is also
in cage three.
Each cage contains a different number of cats.
The orange cat is in cage 2.
There is at least one cat in each cage.
Okay, once we have read the premise,
we can see that this is a grouping game. What are the groups?
What are the elements we are asked to arrange into these groups?
The groups are the three cages that contain the cats. The cats
are the elements we need to arrange. Unlike the previous game,
the elements are not identical. Rather than ten students, we
have seven cats of varying colors.
Our roster might look like this:
Roster: Br, Bl, W, G, T, C, O
Note that we had two colors starting
with B, so we just used Br and Bl to help us differentiate between
those two cats.
Groups: (cages) 1, 2, and 3
Next, let's symbolize the conditions.
The gray cat and the tabby cat are in different cages can be
summarized as:
If G, ^T or if T, ^G.
We can also write this as G ?
T.
The second condition tells us
that the black cat is in cage three only if the white cat is
also in cage three. We can write this as:
3: if B, then W
The next condition tells us that
that each cage contains a different number of cats. This can
be represented as:
1 ? 2 ? 3
We are told that the orange cat
is in cage 2. This is simply represented as:
2: O
Finally, we are told that there
is at least one cat in each cage. We can write this as:
1, 2, 3 ? 0
What else does this statement
tell us? Well, if there are three cages, each containing one
cat, and we have seven cats all together, then the maximum possible
number of cats in any one cage would be five. (Since one cat
would be in each of the other cages.)
Let's draw a diagram to help
us think about the problem.
Is there any information that
we can extract from the conditions and put onto the diagram?
Let's start with the fixed conditions. We can put at least one
line in each cage, since we know there is at least one cat in
each one. We can also place the orange cat (O) in cage 2.
Can we place with certainty any
information from the other conditions? We could also include
the statement about the cats that might be in cage 3 (3: if B,
then W). Though since we don't know yet where any cats (other
than the orange one) are, let's wait on this one until we've
read the question.
Question One
If cage 1 contains only the brown
cat, then which of the following may be true?
(A) Cage 2 contains only the
orange, black, and calico cats.
(B) Cage 3 contains only the black, white, and calico cats.
(C) Cage 2 contains only the tabby, calico, gray, and orange
cats.
(D) The black and white cats are the only two cats in cage 3.
(E) The gray cat is in the same cage as the white cat.
After reading the question, the
first step is to determine whether or not there is any additional
information provided by the question. In this case, we are told
that cage 1 contains only the brown cat, so we can add that to
our diagram. Next we want to consider the original conditions
and determine whether or not in combination with our new information,
we can move anything else to our diagram. In this case, we know
that there are a different number of cats in each cage. Since
there is only one cat is cage 1, we know that there must be at
least two in both cages 2 and 3. Since there are seven cats total,
we also know that we need to place six cats into cages 2 and
3. It cannot be exactly three cats in each cage or that would
violate the rule of there being a different number of cats in
each cage. We can make a note of that at the bottom of groups
2 and 3.
Now let's work through the answer
choices, keeping in mind that we are being asked for which arrangement
may be true. Choices A and B cannot be correct since they place
only three cats in either cage 2 or 3, and we just concluded
that it is impossible to have only three cats in these cages
since that would violate the 1 ? 2 ? 3 rule. Choice C does not
violate the same rule, though when we look over our conditions,
we see that it violates the G ? T rule, so we can eliminate this
option. Choice D puts only the black and white cats in cage 3.
We know that the black cat can only be in this cage if the white
cat is present, which is the other part of this answer choice.
What happens if we add these two cats into our diagram in cage
3? Keep in mind that only the brown one is in cage 1, we already
have the orange one in cage two, and we still need to fit the
other three cats into a cage, and the only one available is two.
When we do this, we can see that
this arrangement will violate the G ? T rule, so this is not
a possibility. Choice E is the only one left, but let's check
it to make sure. The gray cat is in the same cage as the white
one. Is this possible? Let's try placing the two cats into cage
three. We know we need to have the tabby in another cage from
the gray cat, so we must put the tabby in cage two. The other
cats can be in either cage as long as we can figure out an arrangement
that places a different number of cats in each cage. Here is
one possibility, though there are several:
Let's try a second question for
this same problem.
Question Two
If the black cat and the tabby
are two of the cats in cage three, then which of the following
must be false?
(A) The gray cat and the calico
are the only two cats in cage one.
(B) The calico and the brown cat are also in cage three.
(C) There are two cats in cage two.
(D) Besides the black and tabby cats, there are two more cats
in cage three.
(E) The orange cat and the gray cat are in the same cage.
After reading the question, let's
add the new information to our diagram. Remember, we will go
back to our original diagram, not the one on which we added information
specific to the last question.
We know that if the black cat
is in cage three, then the white cat must also be in cage three,
so we can add that information to our diagram.
Now let's work through the answer
choices with the help of our diagram, keeping in mind that we
are looking for the choice that must be false. Starting with
choice A, can we place the gray and calico cats in cage one?
Well, if we have two cats there, we need to have five cats total
in cages two and three. We already know where four of those are
(the black, tabby, and white cats are in cage three, and the
orange cat is in cage two), so we would just need to place the
brown cat. Can we place the brown cat in cage two? No, that would
put two cats in both cages one and two, violating the 1 ? 2 ?
3 rule. Can we put it in cage three? Yes, so this choice is acceptable.
Choice B puts the calico and the brown cat in cage three. If
we add these two cats to the three we have already placed in
cage three, that would give us five cats there, leaving only
two for cages 1 and 2, again violating the 1 ? 2 ? 3 rule. This
choice must be false, which is what we are looking for. Let's
look at the other answer choices, though, just to convince ourselves.
Choice C places two cats in cage two. That would be the orange
cat, plus one other. Can we still place the remaining cats and
satisfy the other conditions? Yes, here is one such arrangement.
Choice D puts a total of four
cats in cage four. Is this possible? Yes, the last example actually
satisfies this condition too. Choice E requires the orange and
gray cats to be in the same cage, and we can easily see that
this is a possible arrangement. Therefore, the one choice that
must be false is choice B.
Summary
Hopefully these sample problems
have helped to illustrate how to create diagrams to enable you
to solve grouping games. As with the other game types, by applying
the strategy, and utilizing your diagram, you will effectively
and quickly be able to solve these games.
Again, here is the strategy for
working through grouping games:
