Which one of the following must be false?

Ravi on January 13, 2019

@Boram,

Happy to help. We're told that a store sells shirts only in small,
medium, and large sizes, and only in red, yellow, and blue colors.
Casey buys three shirts from the store.

_ _ _
1 2 3

Then, we're given the rules.

A shirt type consists of both a size and a color.

_ _ _
_ _ _
1 2 3

Casey does not buy two shirts of the same type.

1) No repeats

Casey does not buy both a small shirt and a large shirt.

2) S - >/L; L - >/S

No small red shirts are available.

3) S - >/R; this means that if S - ->SY or SB

No large blue shirts are available.

4) L - >/B; this means that if L - ->LY or LR

Now let's see if we can make any further inferences for this game
before going to the questions. Because both S and L can only be in two
different combinations each, and because S and L can't appear on the
board together, we know that there must be at least one M on the
board.

M _ _
_ _ _

Let's take a look at the questions now.

Question 1 asks us which answer choice must be false.

Answer A says two of the shirts that Casey buys are small and two are
red. If two of the shirts are small, then we know that SY and SB are
on the board. However, this means that there isn't room for two red
shirts as well, so this answer cannot be true (it must be false). This
is the correct answer choice.

Question 2 says if Casey buys a small blue shirt, which answer choice
must be false?

We know that SB is on the board. We also now know that there ware no
Ls on the board because of rule 2.

Answer A says Casey buys two blue shirts. This could be true, as Casey
could also buy a medium blue shirt, and this wouldn't violate any
rules. This answer is out.

Answer B says Casey buys two red shirts. If we have SB on the board,
then we know that we cannot have any large shirts on the board. We
also know that there are no small red shirts available, so this means
if there are two red shirts, then they both have to be medium red
shirts. However, this would violate rule 1 (no repeats), so this
answer must be false and is our correct answer choice.

Question 3 says if Casey does not buy a medium yellow shirt, which of
the following must be true? Since we know that there has to be at
least one medium on the board, we know that there will be either a
medium red or medium blue shirt on the board.

M _ _
R/B _ _

Now let's look at the answer choices.

Answer A says Casey buys either a medium red shirt or a small blue
shirt. But this doesn't have to be true.

M L L
B R Y

This game board means that A doesn't have to be true, so we can get rid of it.

Answer B says Casey buys either a medium red shirt or a medium blue
shirt. This is exactly what we figured out as soon as we were told
that Casey doesn't buy a medium yellow shirt. Since at least one
medium has to be there, then that one medium has to be either red or
blue. This is the correct answer.

Question 4 says if Casey buys exactly one medium shirt and does not
buy two shirts of the same color, then she cannot buy which one of the
following?

M_ _
_ _ _

We know red, blue, and yellow are all used. We also know that since
there is exactly one M, the other two shirts are either both small or
both large. This is because small and large can't both be on the board
at the same time.

In looking at the colors for S and L, we have

SY or SB
LY or LR

Answer A says a medium red shirt. This would be fine, as we could have

M S S
R Y B

Answer B says a medium yellow shirt.

M L L
Y Y R or

M S S
Y Y B

This wouldn't work because it would force there to be two yellows and
the question says there can't be two shirts with the same color. This
is our answer.

Question 5 says if neither large red shirts nor small blue shirts are
available, which one of the following must Casey buy?

/LR and /SB

This is quite restrictive. We could have one board with all Ms, or one
board with 2 Ms and one S/L slot since S/L can only be one color.

M M M
_ _ _

M M S/L
_ _ Y

Answer A says a red shirt. This doesn't have to be true, as in the
second board Casey could buy MB and MY

M M S/L
B Y Y

Answer B says a medium yellow shirt. This also doesn't have to be
true, as the second board could be

M M S/L
R B Y

Answer C says either a large shirt or a small shirt. This doesn't have
to be true, as the first game board shows (all medium shirts).

Answer D says either a medium red shirt or a medium blue shirt. Casey
must buy one of these. In the first board with all Ms, at least one
has to be red or blue. In the second board, because one slot is
already filled with S or L and Y, R or B must be paired with M because
there cannot be any repeats. Even though we have three colors to
choose from for the two M slots, one of these must be either R or B
so that there aren't repeats.

Even if we put M with Y in the second board, the other M has to be R or B

M M S/L
R/B Y Y

Answer D is our choice for question 5.

Does this make sense? Let us know if you have any more questions!

Julie-V on August 28, 2019

For rule 2, does that mean that she doesn't buy two of the same shirts with the same size AND color? Or does it mean no repeat of colors?

Victoria on August 29, 2019

Hi @Julie-V,

The first rule tells us that a shirt type consists of both size and colour so the second rule means that she doesn't buy two of the same shirts with the same type (i.e. size AND colour).

Hope this helps! Keep up the good work.