An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K— not necessarily in order of the runner’s names as given. The following ordering restrictions apply:

  1. The runner representing K is assigned to lane 4.
  2. Patricia is assigned to the only lane between the lanes of the runners representing F and G.
  3. There are exactly two lanes between Olivia’s lane and the lane of the runner representing G.
  4. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.
  1. Which one of the following is a possible assignment of runners to lanes by the charity they represent?

  1. The lane to which Patricia is assigned must be a lane that is
  1. next to the lane to which Larry is assigned
  2. next to the lane to which Ned is assigned
  3. separated by exactly one lane from the lane to which Ned is assigned
  4. separated by exactly one lane from the lane to which Olivia is assigned
  5. separated by exactly one lane from the lane to which Sonja is assigned
  1. If Olivia is assigned to lane 2, which one of the following assignments must be made? Charity Lane
  1. F 1
  2. G 5
  3. H 1
  4. H 3
  5. J 5
  1. Which one of the following, is a complete and accurate list of runners each of whom could be the runner representing F?
  1. Larry, Ned
  2. Patricia, Sonja
  3. Larry, Ned, Olivia
  4. Larry, Ned, Sonja
  5. Ned, Patricia, Sonja
  1. If Ned is the runner representing J, then it must be true that
  1. the runner representing G is assigned to lane 1
  2. the runner representing H is assigned to lane 2
  3. Larry is the runner representing K
  4. Olivia is the runner representing F
  5. Patricia is the runner representing H
  1. If Larry represents J, which one of the following could be the assignment of runners to lanes?

The Action:

Next up we have runners in five consecutively-numbered lanes, which immediately suggests sequencing, and indeed, we will be concerned with the order of the runners in the five lanes. But we have another concern as well—the runners are also matched up with charities, so in fact the set is a sequencing/matching hybrid. As is usual in hybrid sets, there are several Key Issues to look out for:

1) Which runners must, can, or cannot run in each lane?

2) Which charities must, can, or cannot be represented by the runners in each lane?

3) Which runners must, can, or cannot be matched with which charities?

The Initial Setup:

Since there are two levels to this sequence, one way to set this up is to put the lane numbers on the page with the runners and charities listed off to the side, like so:

With this setup, we have space in our sketch to insert the line of runners, as well as space for the charities the runners represent. As this also makes the matchups of runners to charities readily apparent, both set actions can be accommodated within this one simple sketch.

The Rules:

1) Nice and concrete: We can insert k right into lane 4 in the lower line of our sketch.

2) is slightly complex in that it posits a relationship between runner P and charities f and g.

The use of the word “only” in this rule tells us that no other runner besides P comes between charities f and g, so whether it’s fPg or gPf, this trio will form a solid bloc somewhere in our sketch. Let’s jot this down off to the side for now, and we’ll come back to analyze its implications during Step 4 of the Kaplan Method. Here’s one way to remind ourselves of the rule:

3) Rule 3 also deals with both a runner and charity, in this case runner O and charity g:

4) Here’s a straightforward sequencing rule involving only runners:

N . . . S

is a perfect reminder of this, and we’re ready to combine the rules and see what we can deduce.

Key Deductions:

Now, remember what we said back in Set 2 on this section (and applied in Set 3): One of the best ways to find Key Deductions is to focus on those entities that appear in more than one rule, and here, that entity is charity g. So let’s investigate the possible lanes for charity g. Rule 1 places charity k in lane 4, which severely restricts g’s position: In order to satisfy Rule 2 (that is, to get P between g’s lane and f’s lane), g’s lane can only be lanes 1, 3, or 5—that k in 4 prevents any other possibility. But lane 3 is impossible for g because of Rule 3: If g is in 3, there’s no way to get exactly two lanes between g and runner O. Therefore, charity g must be in either lane 1 or lane 5, and since we know a decent amount about g’s relations to the other entities, it’s worth investigating what happens in both cases.

OPTION 1: If g is in lane 1, then runner P is in lane 2, followed by charity f in lane 3 (Rule 2). Rule 3 kicks in, forcing runner O to lane 4, along with charity k which is always in that lane. That leaves charities h and j to float between lanes 2 and 5, and runners N, S, and free agent L to fill in remaining lanes 1, 3, and 5, so long as N precedes S (Rule 4). This first option, therefore, boils down to this:

You may note that in this option lane 5 must go to runner S or L, as N can’t be in 5. You can write this in if you like, or simply deal with the remaining runners accordingly in each question.

OPTION 2: If g is in lane 5, then runner P is in lane 4 (along with k), and charity f takes lane 3, again thanks to Rule 2. Rule 3 forces O into lane 2. Again, charities h and j will float, this time between lanes 1 and 2. And again, runners N, S, and L will take lanes 1, 3 and 5, with N before S in 1 or 3 and S or L in 5. Here’s this second option:

Notice that charity f is in lane 3 in both options, which means that this is another solid deduction that must hold for the entire set.

The Final Visualization now consists of the two options on the page. Every acceptable ordering and matchup will have to correspond to one of these two options, which will lighten our workload considerably when it comes to answering the questions.

The Questions:

1. (E)

Acceptability Q. 1 is our first opportunity to use the options we created—note that we’re concerned only with charities here. We deduced that in both options, charity f must be in lane 3, which allows us to quickly eliminate (A), (B), and (D). Of the remaining choices, (C) violates Rule 1—k must be in lane 4—which is why the ordering in (C) doesn’t jibe with either of the options listed on the page. (E) remains and gets the point for 19.

2. (D)

In Option 1, P is in lane 2 and O is in lane 4. In Option 2, they’re reversed: O is in 2 and P is in 4. Either way, (D) has it right: P and O must be separated by exactly one lane.

(A) In Option 1, P is in lane 2 and L can take lane 5.

(B), (C) In Option 2, P is in lane 4 and N can be in lane 1. So contrary to both (B) and (C), it’s possible to have exactly two lanes between the lanes of P and N.

(E) is impossible: In Option 1, S is either right next to P or separated by two lanes from P; in Option 2, S, in lanes 3 or 5, is always right next to P.

3. (B)

Here’s our first hypothetical, and it steers us clearly toward Option 2, the only option in which O is in lane 2. All we need to do is check the choices against Option 2, looking for the assignment that must be made. Charity g in lane 5 is the one the testmakers chose, choice (B).

(A) and (D) are impossible—charity f must always be in lane 3.

(C) is possible only; in Option 2, h could be in lane 1, but could also be in lane 2, with j in 1.

(E) No—in Option 2, charity j must be in either lane 1 or 2.

4. (D)

Rule 2 directly rules out Patricia as f’s representative, as Patricia must run in the lane between the lanes of charities f and g. That allows us to kill (B) and (E) right off the bat. Both Larry and Ned appear in all of the remaining choices, so those runners must be part of the correct list, and we need not even bother with them. The question then boils down to Olivia and Sonja. If neither can represent f, then (A) is it. If only O can, then we choose (C), and if Sonja can, (D)’s the winner. A glance at either option on the page confirms that Sonja can represent f, a charity that we know is always in lane 3.

5. (B)

For the sake of comparison, let’s approach Q. 23 without the benefit of the limited options worked out above. Ned is to represent charity j, and we need to determine what must be true. Well, for all we know, there are lots of ways this can happen, and as there’s not much information on Ned (only Rule 4, which doesn’t narrow things down much here), and no information on charity j (one of our “free agents” in this set), we have no choice but to pop N and j into lanes and see how they fare.

What about lane 1? If N represents j in lane 1, nothing is triggered directly, so we better consider our large bloc of entities, the fPg bloc. With charity k always in lane 4, and j now in lane 1, the only way to space charities f and g with exactly one lane between them for P would be to place them in lanes 3 and 5, with P in lane 4. But f and g could theoretically go in either order, so which is in 3 and which is in 5? Just have to try it out, in which case we find that g in 3 is impossible, because there would be no way to satisfy Rule 3. (The more perceptive test-takers might realize at this point that g can never be in 3, but others who proceed piecemeal like this may not realize this and think that this observation only applies to this question, thereby diminishing the value of this important insight.) So in this case, with N representing j in lane 1, charity f must be in 3 and charity g in 5, which forces runner O and the one remaining charity, h, into lane 2.

Whew! Are we done? No; all we’ve done is prove that N can represent j in lane 1, which would help us in a “could be true question,” but is only part of the battle in a “must be true” question. For all we know, N and j can occupy other lanes, so we now have to test out other scenarios. What about N and j in lane 2? The only way to place the fPg bloc would be to place them again in lanes 3, 4, and 5, respectively, but this leads to a violation: Rule 3 would then force O into lane 2, but we’ve just placed N there.

Moving along, could N and j take lane 3? No, because there’s no way to then place the fPg bloc at all. Lane 4 is off limits, because charity k already occupies that space, and lane 5 is no good, because N can never be in lane 5, lest it violate Rule 4.

So after this somewhat lengthy analysis, we’ve proven that there’s only one lane in which N can represent charity j, and it leads to this scenario:

Now we’re in a position to recognize for sure what must be true—charity h’s runner is assigned to lane 2, choice (B). All of the other choices here must be false:

(A) No, charity g’s runner is assigned to lane 5.

(C) No, Patricia represents charity k here.

(D), (E) No, Olivia represents charity h here.

5. (A)

Finally, we’re looking for an acceptable ordering of runners if Larry is matched with charity j. Our options help us immediately: No matter what, O and P must take lanes 2 and 4, in either order. But three of the five choices—(B), (C), and (E)—don’t conform to this, and can be axed on this basis alone. That leaves (A) and (D), both acceptable orderings in general, but obviously one will not accord with the mandate in the stem. Both have O in lane 2 and P in lane 4, which places this scenario squarely in Option 2. In that Option, charity j is either in lane 1 or lane 2, and if L is to be matched with j, it must be in lane 1, leaving N and S to lanes 3 and 5, respectively (Rule 4). L O N P S works, and therefore could be the assignment of runners to lanes under these conditions.

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