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Section One Logical Reasoning
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Question #401
From among ten stones, a jeweler will select six, one for each of six rings. Of the stones, threeF, G, and Hare
rubies; threeJ, K, and Mare sapphires; and fourW, X, Y, and Zare topazes. The selection of stones must meet the
following restrictions:
At least two of the topazes are selected.
If exactly two of the sapphires are selected, exactly one of the rubies is selected.
If W is selected, neither H nor Z is selected.
If M is selected, W is also selected.
Which one of the following could be the selection of stones?
A. F, G, H, M, X, Y
B. F, G, J, K, M, W
C. F, G, J, K, W, X
D. G, H, J, X, Y, Z
E. G, H, K, W, X, Z
Answer: D
Even if you were blown away by this games rules, this is a point worth getting, and worth getting quickly. Rule 1s
requirement of 2 topazes minimum is violated by
B.. Rule 2s 2 sapphires : 1 ruby ratio is violated by C., which gives us 2 : 2 : 2. Rule 3 is violated by E., which
gives us W, H, and Z all at once. And Rule 4 is violated by A., which gives us M without W. D. must be correct
sight unseen, and perhaps finding it, and then analyzing it, can provide us with some needed insight into the game
and the heart to keep going with it.
Question #402
From among ten stones, a jeweler will select six, one for each of six rings. Of the stones, threeF, G, and Hare
rubies; threeJ, K, and Mare sapphires; and fourW, X, Y, and Zare topazes. The selection of stones must meet the
following restrictions:
At least two of the topazes are selected.
If exactly two of the sapphires are selected, exactly one of the rubies is selected.
If W is selected, neither H nor Z is selected.
If M is selected, W is also selected.
Which one of the following must be true?
A. G is selected.
B. J is selected.
C. X is selected.
D. Of at least one of the three types of stones,exactly one stone is selected.
E. Of at least one of the three types of stones,exactly three stones are selected.
Answer: E
Those who worked out the number options we laid out above had no trouble with this one. No matter how you
slice it, at least one stone typeand possibly two are represented by exactly 3 stones. And even if you didnt work out
every possibility, simply seeing that 2 : 2 : 2 was an impossible ratio(because of Rule 2) might have led you to
suspect that E. was a true statement. A., B., and C. are far from necessarily truethere are lots of possibilities in
which G, J, and X respectively are not present. And the possibility of choosing, say, 3 sapphires + 3 topazes means
that D. need not be true either.
Question #403
From among ten stones, a jeweler will select six, one for each of six rings. Of the stones, threeF, G, and Hare
rubies; threeJ, K, and Mare sapphires; and fourW, X, Y, and Zare topazes. The selection of stones must meet the
following restrictions:
At least two of the topazes are selected.
If exactly two of the sapphires are selected, exactly one of the rubies is selected.
If W is selected, neither H nor Z is selected.
If M is selected, W is also selected.
If Z is selected, which one of the following could be true?
A. All three of the sapphires are selected.
B. Both J and M are selected.
C. Both K and M are selected.
D. None of the rubies is selected.
E. None of the sapphires is selected.
Answer: E
Write out your roster, circle Z on it, and realize that thats a trigger for rejecting W (Rule 3s contrapositive) and thus
for rejecting M (Rule 4s contrapositive). Thats all we need. With sapphire M gone, clearly A.,B., and C. are
impossible. And if D. were trueno rubiesour remaining 2 sapphires and maximum of 3 topazes would leave us short
of our desired total of 6. So E. is possible by default. Indeed, all three rubies plus topazes X, Y, Z would constitute
an acceptable set of 6.
Question #404
From among ten stones, a jeweler will select six, one for each of six rings. Of the stones, threeF, G, and Hare
rubies; threeJ, K, and Mare sapphires; and fourW, X, Y, and Zare topazes. The selection of stones must meet the
following restrictions:
At least two of the topazes are selected.
If exactly two of the sapphires are selected, exactly one of the rubies is selected.
If W is selected, neither H nor Z is selected.
If M is selected, W is also selected.
If exactly two rubies are selected, which one of the following must be true?
A. H is selected.
B. J is selected.
C. Z is selected.
D. Exactly one sapphire is selected.
E. Exactly two topazes are selected
Answer: D
This is sheer arithmetic. If we choose exactly 2 rubies, we cannot choose 2 topazesremember, 2: 2 : 2 is
forbiddenso we need exactly 3 topazes (and have proved thatE.is impossible, incidentally).And 2 rubies + 3 topazes
only = 5 stones, so there will be exactly 1 sapphire chosen, choice D.What that sapphire is, is unclear; it could be J,
as B. suggests, but need not. And A. and C. are much more certain as to the identity of particular stones than we
can possibly be, under this questions circumstances.
Question #405
From among ten stones, a jeweler will select six, one for each of six rings. Of the stones, threeF, G, and Hare
rubies; threeJ, K, and Mare sapphires; and fourW, X, Y, and Zare topazes. The selection of stones must meet the
following restrictions:
At least two of the topazes are selected.
If exactly two of the sapphires are selected, exactly one of the rubies is selected.
If W is selected, neither H nor Z is selected.
If M is selected, W is also selected.
Which one of the following must be true?
A. The selection of stones includes at least one ruby.
B. The selection of stones includes at most two rubies.
C. The selection of stones includes either F or Z, or both.
D. The selection of stones includes either X or Y, or both.
E. The selection of stones includes either X or Z, or both.
Answer: D
We have deduced that at least two topazes must be chosen at all times. And which two? Hard to say; but since we
can never choose W and Z together (Rule 3), we will always need to choose at least one of the remaining topazes.
Thats all that D. is saying. Our work in Step 4, above, demonstrates that neither A. nor B. need be true. Contrary to
both A. and B., we could choose no rubies at all (if we used 3 sapphires and 3 topazes). The remaining choices may
require a little trial and error, but the selection of, for instance, "G,J,K,M,W,Y" proves that neither C. nor E. need
be true.
Question #406
From among ten stones, a jeweler will select six, one for each of six rings. Of the stones, threeF, G, and Hare
rubies; threeJ, K, and Mare sapphires; and fourW, X, Y, and Zare topazes. The selection of stones must meet the
following restrictions:
At least two of the topazes are selected.
If exactly two of the sapphires are selected, exactly one of the rubies is selected.
If W is selected, neither H nor Z is selected.
If M is selected, W is also selected.
If J and M are the only sapphires selected, which one of the following could be true?
A. F and G are both selected.
B. F and X are both selected.
C. G and H are both selected.
D. G and K are both selected.
E. Y and Z are both selected.
Answer: B
Interestinglyif frustratinglythis last question proves to be one of the most concrete, and hence one of the easiest.
We hope that you didnt get bogged down on some of the earlier ones and miss out onthis one. In your roster, select
J and M, and cross out K, in line with the question stems if. Rule 4 requires that you select
W, and Rule 3 requires that you cross out H and Z. At the same time, Rule 2 (we use it at last!) kicks inthere will
be exactly 1 ruby chosen, and hence 3 sapphires. And with Z gone, those 3sapphires are W, X, and Y. So: J, M,
W, X, and Y are definitely chosen, and the final stone will be a ruby, either F or G.
B. is therefore the only possibility among the choices. A. would stick us with a total of 7 stones. And each of the
remaining choices mentions a stone that we have definitively rejected: H in C., K in D.,and Z in E.
Question #407
There are exactly ten stores and no other buildings on Oak Street. On the north side of the street, from west to east,
are stores 1, 3, 5, 7, and 9; on the south side of the street, also from west to east, are stores 2, 4, 6, 8, and 10. The
stores on the north side are located directly across the street from those on the south side, facing each other in pairs,
as follows: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10. Each store is decorated with lights in exactly one of the
following colors: green, red, and yellow. The stores have been decorated with lights according to the following
conditions:
No store is decorated with lights of the same color as those of any store adjacent to it.
No store is decorated with lights of the same color as those of the store directly across the street from it.
Yellow lights decorate exactly one store on each side of the street.
Red lights decorate store 4.
Yellow lights decorate store 5.
Which one of the following could be an accurate list of the colors of the lights that decorate stores 2, 4, 6, 8, and
10, respectively?
A. green, red, green, red, green
B. green, red, green, yellow, red
C. green, red, yellow, red, green
D. yellow, green, red, green, red
E. yellow, red, green, red, yellow
Answer: B
The right answer is an acceptable matching for the five south side streets, and weve got most of that worked out.
Stores 4 and 6 are red and green respectively, so C. and D. can be crossed out. There is one and only one yellow
light, so A. with no yellows and E. with two are both eliminated. B. is all thats left.
Question #408
There are exactly ten stores and no other buildings on Oak Street. On the north side of the street, from west to east,
are stores 1, 3, 5, 7, and 9; on the south side of the street, also from west to east, are stores 2, 4, 6, 8, and 10. The
stores on the north side are located directly across the street from those on the south side, facing each other in pairs,
as follows: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10. Each store is decorated with lights in exactly one of the
following colors: green, red, and yellow. The stores have been decorated with lights according to the following
conditions:
No store is decorated with lights of the same color as those of any store adjacent to it.
No store is decorated with lights of the same color as those of the store directly across the street from it.
Yellow lights decorate exactly one store on each side of the street.
Red lights decorate store 4.
Yellow lights decorate store 5.
If green lights decorate store 7, then each of the following statements could be false EXCEPT:
A. Green lights decorate store 2.
B. Green lights decorate store 10.
C. Red lights decorate store 8.
D. Red lights decorate store 9.
E. Yellow lights decorate store 2.
Answer: D
A mini-sketch for this question should recopy what we know and add "G" to store 7. Store 9, which we knew was
"G/R," is now definitively "R." And that’s it; we cannot confirm anything more on the south side. But thats
enough: store 9 has to be red, so D. cannot be false.
Of A. and E., exactly one is truestore 2 is green or yellowbut we dont know which. Store 8could be red or yellow,
so C. can be false. And store 10 remains uncertain, so B.can be false too.
Question #409
There are exactly ten stores and no other buildings on Oak Street. On the north side of the street, from west to east,
are stores 1, 3, 5, 7, and 9; on the south side of the street, also from west to east, are stores 2, 4, 6, 8, and 10. The
stores on the north side are located directly across the street from those on the south side, facing each other in pairs,
as follows: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10. Each store is decorated with lights in exactly one of the
following colors: green, red, and yellow. The stores have been decorated with lights according to the following
conditions:
No store is decorated with lights of the same color as those of any store adjacent to it.
No store is decorated with lights of the same color as those of the store directly across the street from it.
Yellow lights decorate exactly one store on each side of the street.
Red lights decorate store 4.
Yellow lights decorate store 5.
Which one of the following statements must be true?
A. Green lights decorate store 10.
B. Red lights decorate store 1.
C. Red lights decorate store 8.
D. Yellow lights decorate store 8.
E. Yellow lights decorate store 10.
Answer: B
This one is a slam-dunk. We have deduced that store 1 has red lights.
Question #410
There are exactly ten stores and no other buildings on Oak Street. On the north side of the street, from west to east,
are stores 1, 3, 5, 7, and 9; on the south side of the street, also from west to east, are stores 2, 4, 6, 8, and 10. The
stores on the north side are located directly across the street from those on the south side, facing each other in pairs,
as follows: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10. Each store is decorated with lights in exactly one of the
following colors: green, red, and yellow. The stores have been decorated with lights according to the following
conditions:
No store is decorated with lights of the same color as those of any store adjacent to it.
No store is decorated with lights of the same color as those of the store directly across the street from it.
Yellow lights decorate exactly one store on each side of the street.
Red lights decorate store 4.
Yellow lights decorate store 5.
If green lights decorate five stores on the street, then which one of the following statements must be true?
A. Green lights decorate store 9.
B. Red lights decorate store 2.
C. Red lights decorate store 7.
D. Red lights decorate store 10.
E. Yellow lights decorate store 8.
Answer: E
The trick here is to figure out which of the ten stores are the five that will have green lights. On the north side, with
store 3 always green, we will have to have a green light in store 7 or 9. (Remember, adjacent greens are
prohibited.) It may be smart to draw out both possibilities. And happily the south side is determinable.
On the south side, we need three more greens, and clearly theyll have to be stores 2, 6, and10. And with store 4
always red, the requisite (Rule 3) yellow will go to store 8. So here are the two options:
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