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The Action

Arranging seven people and two empty chairs in a circle. The key issues on which the questions will be based are:

- Who can sit next to whom?

- Who must sit next to whom?

- How far away are some entities from other entities?

- Where are the entities with respect to the empty chairs?

The Initial Setup

Anytime you have an odd number of entities to place, you can’t imagine a standard circle sectioned off like pieces of a pie, because this will always come out even. So instead, try drawing nine dots spaced out in a circle, and number the seats #1 through #9, making #1 at the top and working clockwise for #2-9.

(Note: Assigning actual seat numbers is, in fact, a step beyond what you would normally do in this type of game-they’re included solely as convenient reference points for the following explanations.)

Since this game specifies how people will be seated in terms of being to the left or right of one another, we must also take into account which way is left and right. In circle sequence games, people always sit facing the centre of the circle; those sitting at the top of the circle, therefore, will have their left and right reversed from those on the bottom, who perceive left and right as you do. (Try to picture yourself sitting at the top of the circle, facing the centre, and again at the bottom; you’ll see how left and right reverse themselves.) Make a mental note that left always points clockwise and right always points counterclockwise.

The Rules

1. With any circle sequencing game, we start by picking one of the entities and placing him/her at some point on the circle. Since we see S mentioned in most of the rules, we will start by placing her in a seat, let’s say #1, with M to her immediate right in seat #9 (remember, right is counterclockwise).

2. Since Sybil’s already placed, rule 2 is a good one to incorporate next. Counting off three chairs to the right of Sybil (remember-counterclockwise) places R in chair #7.

3. This means that seats #2, #6, and #8 can’t be empty, so remember this.

4. If L and O sit immediately next to each other, we need to locate a pair of adjacent chairs for them.

The possible candidates are: #2-3, #3-4, #4# and 5-C. #3-4 and #4-5 won’t work since each those cases would necessitate placing an empty chair next to Randi or Sybil, which violates the previous rule. So, we know that the L-O pair will sit either seats #2-3 or seats #5-6. Note that even determine which of those pairs L and O actual in, we still don’t know which person will sit which chair.

Key Deductions

If L and O sit in #2-3, the two empty chairs must be #4 and #5. Similarly, if L and O are in #5-6, the empties must be #3--4. In any event, we can now ascertain that chair #4 MUST be empty. Also, in either case one of the two remaining people, P or N, must sit in chair #8.

The Final Visualisation

The placements of S, M, and R are determined and; I and O are restricted to two pairs of chairs; the empty chairs are next to each other in one of two ways (#3-4, or #4-5); and N and P, the people not mentioned in any rules, are left to fill in the spaces as necessary.

The Big Picture

- In circle sequence games, always begin by placing the most prominent entity (the one connected to the most rules) into the circle first, and then build around it.

- It doesn’t matter where in the sketch you place the first entity; the relationships between the entities is the crucial thing, which will remain consistent whether you begin by placing S in seat #1 or seat #6, for example.

- Check to see if the game cares about left and right. Some games will simply say “next to” and “across from.” Others, like this one, care about direction.

- If left/right is an issue, make certain which way is left and which way is right from any chair. If the entities are facing inward, as is usually the case, right will be counterclockwise and left will be clockwise.

The Questions

19. (D) As mentioned earlier, a key deduction answers this one. We deduced that the two empty chairs must be either chairs #3 and #4, or chairs #4 and #5. Either way, the two empties are adjacent.

Neither (A) nor (B) needs to be true, as Pam and Naomi could sit in chair #2. Although L could sit next to an empty chair, L in #6 and O in #5 proves that L doesn’t have to, so (C) is out. The same example shows that (E) isn’t necessarily so.

Remember:

- When the first question offers no hypothetical information, the answer will be something that will hold true for the entire game. So, if you hadn’t already picked it out as part of the key deductions, build it into your visualisation and treat it as a valid rule for the rest of the game.

20. (E) This would place O in seat #5, and L next to O in seat #6 (rule 4). The empties must therefore take seats #3 and #4. The only uncertainty remaining is that N and P are free to float between seats #2 and #8. The answer is derived from this fact. While Pam could sit in seat #2, next to Sybil, she could also sit in seat #8.

(A) must be true, since three seats away from R is empty seat #4. Furthermore, O, in seat #5, is next to empty chair #4, so (B) can’t be false, either. We can rule out (C) since in this case L does sit next to R. Finally, (D) gets thrown out because regardless of which of N and P sits in #2 and which sits in #8, there must be exactly two people (S and M) between them.

Remember:

- When questions look for something that could be false, it’s a good bet that the answer will involve one of the more ambiguous entities. In this case, Pam and Naomi were the only unplaced people, which quickly narrowed down the choices to (D) and (E).

21. (A) With no new hypothetical information to go on in this “cannot be true” question, we’re left to test each choice by trying to make each one true. When we find the one that can’t be made to work, then that’s our winner. Luckily, the answer comes immediately in choice (A): if L and O take seats #2 and #3, Pam must sit in either #6 or #8, next to Randi. If the other scenario exists (L and O in #5 and #6), then Pam must sit in seat #2, next to Sybil, or seat #8, next to Randi.

By placing O in seat #3 or #5, we can make (B) true. The fact that P can sit between R and M rules out (C). (D) and (E) are out as well, since seat #2 can be occupied by L or N.

Remember:

- If you can’t decide where to begin in a question or to test out a choice, work with the entities that take up the most space-in this case, the L/O combo. When major entities are limited to two possibilities, simply try each and see what happens.

22. (C) Lisa can sit next to an empty chair if she sits in seat #3, with O in #2, or if she sits in seat #5, with O in #6. In the first case, P and N must sit in seats #6 and #8, in either order. So there are actually two possible arrangements attached to the first case. The same holds true for the second case, the scenario with L and O in #5 and #6, respectively. P and N could then float between seats #2 and #8. So, there are two general ways to place Lisa next to an empty seat, each containing its own two variations, resulting in four different possible arrangements overall.

Remember:

- When asked for the number of different possible arrangements, simply try each variation of the seating until you’ve exhausted all possibilities.

23. (C) In order to determine the exact locations of everyone, we need to resolve the

The intuitive approach to this problem, therefore is to look for a choice that mentions the people in each of these pairs. (A), (C), and (E) look like good candidates, so let’s begin with those. If Pam sits in seat #8, we can take care of (A) in two ways: L in #2 or L in #5. (E) can be discarded in the same way, by simply substituting O for L and N for P (C), however, does the trick: the only way to get exactly one person between L and N is if N sit in #8 and L sits in #6. This forces O into #5, P into #2, with #3 and #4 left empty. As for (B) and (D) which we never had to check after all, neither one allows for a definitive placement of the two perpetual free agents, N and P.

Remember:

- When looking for a statement that will precisely determine the placement of every entity, scan the list of choices for ones that deals with the game’s most ambiguous characters, and to those out first.

24. (E) The best approach here is to create a fresh sketch for this question (it doesn’t take long at all), and reapply the rules. Continue to keep S in seat # 1. Now, instead of being in #9, M. will be in seat #3. Following original rule 2, R remains in seat #7. Reapplying rule 3 tells us that now seats #2, #6, #8, and #9 can’t be empty, so the only remaining seats, #4 and #5, must be the empties. Finally, applying rule 4 tells us that L and O must share the only two remaining adjacent seats, #8 and #9 (in either order), while N and P are left to float between #2 and #6.

Armed with all of this information, it’s easy to see that (A) through (D) can all be true (choice (A) in fact must be true), while choice (E) can’t possibly be true. The chairs that are three chairs away from Sybil are #4, which is empty, and #7, which is Randi’s.

Remember:

- If you’re pressed for time, you may wish to skip a question that involves a change in the original rules, especially one that pretty much requires you to rethink the rules from scratch.