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Choosing six out of eight people to play basketball, and breaking up the six players into two teams of three players each. The unusual twist on this ‘`grouping game of distribution” (meaning that it’s a game in which you’re putting the entities into subgroups), is that each player on a team plays opposite a specific player on the other team. Here are the key issues that you should anticipate the questions will be based on:

- Which players are chosen and which ones sit on the bench?

- Which players can or must play together?

- Which players cannot play together?

- Which players can and cannot play opposite each other?

The Initial Setup

If you were the coach for this situation in real life, how might you use pencil and paper to keep track of it all? Here’s a simple, intuitive way:

The Rules

1. Rule 1 simply breaks the players into two groups: C, E, K, and P are tall, and the others-G, J, M, and T-are short.

2. Thanks to rule 1, we can now get a little more specific about rule 2. C, E, K, and P, the tall players, can play opposite each other only. Let’s analyse what this means in practical terms-terms that will help you answer the questions. Here are the implications of this rule that make the game much easier to handle.

- What would violate this rule? Answer: a tall player matched up with a short player. If that’s unclear, form an example to test your understanding of the rule.

- If two of the tall players, Charles and Edna, for example, are chosen and placed on one team, then the other two, Katya and Paulina, have to be chosen and placed on the other team.

- There must be either two or four tall players chosen. There’s no way to match them up with each other if an odd number is chosen. (This thought leads to a bigger deduction regarding the overall number breakdown, which you’ll see below under “Key Deduction.” Did you see it on your own?)

- A team cannot be made up of three tall players, because there would have to be three tall players for the other team, but there are only four tall players total.

- Likewise, a team cannot be made up of three short players. The tall player or players on the other team would have no tall players to oppose.

3. Rule 2 further clarifies the tall-people issue. If Paulina is chosen to play, then Charles is chosen and plays opposite her (inferably on the other team). This is straightforward enough. Just don’t make the lethal mistake of inferring that the reverse is true, because it isn’t. If Charles is chosen, Paulina can, but does not have to, play opposite him.

4. This one is similar to the previous one. If Terence is chosen, Charles is chosen but plays on the same team as Terence. Once again, the rule can’t go backwards. Choosing Charles implies nothing about choosing Terence.

Key Deduction

The key deduction in this game involves the breakdown of tall and short players chosen. One of the implications of rule 2 is that there has to be either two or four tall players chosen. Taken one step further, this means that the coach has to choose either all four tall players and two short players, or all four short players and two tall players. One more deduction stems from this, but it’s not as obvious, and therefore it’s not a tragedy if you don’t see it up front. In fact, it’s tested for specifically in question 1, so we’ll cover it there.

The Final Visualisation

- One way to latch onto the action of a game is to place the game in a real-life context. How would you handle this if you were the coach? Break down the mystique. If you approach the games as ordinary, everyday situations, they won’t be as baffling and you’ll be able to think through them more clearly.

- When the entities are broken down into two distinct groups, you may find it helpful to differentiate them by using capital and lowercase letters.

- A good way to grasp the meaning of a rule is to ask yourself, “What would violate this rule?” Take a second, if it helps, to try out a concrete example that tests your understanding of the rule. Otherwise the rule will continue to be abstract, unclear, unhelpful.

- When a game involves numbers, always see if you can deduce specific number breakdowns. Keeping these firmly in mind gives you a distinct advantage when it comes to working through the questions.

The Questions

1. (D) Given no new information, we’re asked for the player who must be chosen. All the Vies about Charles make him a likely suspect, and in fact, Charles is the one. You can get this by working off of the key deduction. If the coach chooses all four tall players and two short players, then obviously Charles, a tall player, must be chosen. If, on the other hand, the coach chooses all four short players and only two tall players, then short Terence must be chosen, and Charles as well, thanks to rule 4. The big deduction pays off - there are only two ways to combine tall and short players to get the required six, and Charles is chosen in both of them.

Edna, choice (A), and Katya, choice (C), can both be bench warmers if Charles and Paulina are chosen along with all four short players. However, Terence, choice (B), and Jerrod, choice (E), can sit out if Greg and Mort play along with all four tall players.

Remember:

- When a question offers no hypothetical information (no if-clause), it means that it must be

possible to deduce the answer from the rules alone. Build the new piece of information (in this case, the fact that Charles must be chosen) into your global view of the game.

2. (D) We saw earlier from the implications of rule 2 that choice (C), three short players on one team, and choice (E), three tall players on one team, are both impossible. Both cases result in at least one tall player not being matched up against another tall player. (A) violates rule 3-Paulina and Charles can never be on the same team. Choice (B) is a little more subtle, but the key deduction comes in handy to help us see why tall Charles, short Jerrod, and short Mort cannot form a team. If they were a team, then the four-short-andtwo- tall scenario would be in effect, placing Terence and Greg on the other team. But that contradicts rule 4, so it’s no good. The players in the remaining choice, choice (D), form an acceptable team; the opposing team would have to include Charles, Jerrod, and Terence.

Remember:

- When facing an acceptability question, simply check the choices against each rule, eliminating the ones that violate the rules until you’re left with only one choice. If you do your work carefully, there’s no need to check the remaining choice; just circle it and move on.

3. (B) The shortcut that answers this question stems from our key deduction. There must be four tall players chosen, and two short to round out the roster, or four short players chosen with two tall to complete the teams. Therefore, the two bench players must be in the same category: If one were tall and the other short, then the 4-2 split would be impossible to achieve. Specifically, choice (B) is the only one that places one tall and one short player on the bench, leaving three tall and three short players for the game, which we’ve already deduced is an impossibility.

Remember:

- You can often find a shortcut by focusing on the categories that the entities fall into rather than the entities themselves. Notice that we quickly answered this question by finding a choice that includes one tall player and one short player, rather than working out possible lineups for each choice.

4. (D) Paulina can’t play on the same team as Terence, because according to rule 4, Charles would have to be on that team-but that would violate rule 3, which states that Paulina must play opposite Charles. All of the players in the other choices make fine teammates for Paulina.

Remember:

- In questions like this, scan the list of choices for entities that jump out, ones that seem

suspicious. Terence is a good bet because he has a whole rule dedicated to him, which by nature restricts him more than the players in the other choices.

5. (C) According to rule 2, if Katya and Charles, two tall players, are one team, Paulina and Edna must play on the other team. The third player on that team can’t be Terence (rule 4), but could be any of the other three short players, giving us PEg, PEj, and PEm as possibilities. The correct answer is therefore 3, choice (C).

6. (E) is correct.

(A) Paulina-Greg-Jerrod versus Charles-Trent-Mort. Bench: Edna and Katya. Could be true-cross it off:

(B), (C) Paulina-Katya-Mort versus Charles-Edna-Terence. Bench: Greg and Jerrod. Could be true-cross them both off:

(D) Charles-Mort-Terence versus Karya-Greg-Jerrod. Bench: Paulina and Edna. Could be true-cross it off.

(E) This is the one that’s impossible. If Terence is chosen, then Charles must play on his team, resulting in a team of Terence, Edna, and Charles. Thanks to rule 2, the other tall players Paulina and Katya, must play on the other to team oppose Edna and Charles, so it’s impossible for Katya. to sit on the bench.

Remember:

- In questions that offer no hypothetical information, sometimes the only way to proceed is to work out each choice, which could be time consuming. A question like this is therefore a good candidate to skip and come back to later if time permits.

- You can eliminate an answer choice to a “cannot be true” question simply by coming up with one example in which the choice is true.