This is Section 1, Question 18.
There are exactly three recycling centers in Rivertown: Center 1, Center 2, and Center 3. Exactly five kinds of material are recycled at these recycling centers: glass, newsprint, plastic, tin, and wood. Each recycling center recycles at least two but no more than three of these kinds of material. The following conditions must hold:
Any recycling center that recycles wood also recycles newsprint.
Every kind of material that Center 2 recycles is also recycled at Center 1.
Only one of the recycling centers recycles plastic, and that recycling center does not recycle glass.
First, we know that there are 3 sections and that each recycles at least 2 but no more than 3.
1 _ _ (_)
2 _ _ (_)
3 _ _ (_)
The options are glass, newsprint, plastic, tin, and wood:
G N P T W
Rule 1: If it recycles W it also recycles N, so:
W -> N & N -> W (The latter holds to be true since it is called a contrapositive).
Rule 2: If in Center 2 then also in Center 1, so:
2 -> 1 & 1->2 (Again, this is the contrapositive)
This rule has a hidden point as well, it states that if Center 2 recycles three materials, Center 1 will recycle those three as well. If Center 2 recycles 2 materials, Center 1 will recycle those 2 as well. Anything 2 has, 1 does as well.
Rule 3: Only one P, and that center does not recycle G. This allows us to infer that P cannot be in Center 2 because of Rule 2.
Let’s try question 18:
Which one of the following could be an accurate account of all the kinds of material recycled at each recycling center in Rivertown?
(A) Center 1: newsprint, plastic, wood; Center 2: newsprint, wood; Center 3: glass, tin, wood
(B) Center 1: glass, newsprint, tin; Center 2: glass, newsprint, tin; Center 3: newsprint, plastic, wood
(C) Center 1: glass, newsprint, wood; Center 2: glass, newsprint, tin; Center 3: plastic, tin
(D) Center 1: glass, plastic, tin; Center 2: glass, tin; Center 3: newsprint, wood
(E) Center 1: newsprint, plastic, wood; Center 2: newsprint, plastic, wood; Center 3: glass, newsprint, tin
We aren’t given any new rules here, so we will just use the rules we have already pulled out.
Rule #1: If they recycle W they must have N too. Option (a) does not have both, we eliminate this. Going through the rest we know that no other options violate rule 1, so let’s move on to rule 2.
Rule #2: If it’s in center 2, it must also be in center 1. Option (c) is missing tin in center 1, so we eliminate this. Going through the rest we do not see any other violations of this rule, so let’s move on to rule 3.
Rule #3: Only one center can recycle plastic and that one cannot have glass. We see that in option (d) both glass and plastic is being recycled in the same center, so we can cross this off. We also see that option (e) violates this as two centers are recycling plastic. We are now left with (b), which is our correct answer.