LRDI Practice Set 9

Three couples—John and Kate, Lewis and Marie, and Nat and Olive have dinner in a restaurant together. Kate, Marie, and Olive are women; the other three are men.

Each person orders one and only one of the following kinds of entrees: pork chops, roast beef, swordfish, tilefish, veal cutlet. The six people order in a manner consistent with the following conditions:

  1. The two people in each couple do not order the same kind of entree as each other.
  2. None of the men orders the same kind of entree as any of the other men.
  3. Marie orders swordfish.
  4. Neither John nor Nat orders a fish entree.
  5. Olive orders roast beef.
  1. Which one of the following is a complete and accurate list of the entrees any one of which Lewis could order?
  1. pork chops, roast beef
  2. pork chops, veal cutlet
  3. pork chops, swordfish, veal cutlet
  4. pork chops, roast beef, tilefish, veal cutlet
  5. pork chops, roast beef, swordfish, tilefish, veal cutlet
  1. Which one of the following statements could be true?
  1. John orders the same kind of entree as Marie does.
  2. Kate orders the same kind of entree as Nat does.
  3. Lewis orders the same Kind of entree as Nat does.
  4. Marie orders the same kind of entree as Olive does.
  5. Nat orders the same kind of entree as Olive does.
  1. Which one of the following statements must be true?
  1. One of the men orders pork chops or veal cutlet.
  2. One of the men orders swordfish or veal cutlet.
  3. Two of the women order tilefish.
  4. None of the men orders a fish entree.
  5. Exactly one of the women orders a fish entree.
  1. If John orders veal cutlet, then which one of the following statements must be true?
  1. Kate orders roast beef.
  2. Kate orders swordfish.
  3. Lewis orders tilefish.
  4. Lewis orders veal cutlet.
  5. Nat orders pork chops.
  1. If none of the six people orders pork chops, then which one of the following statements must be true?
  1. John orders veal cutlet.
  2. Kate orders tilefish.
  3. Lewis orders tilefish.
  4. One of the men orders swordfish.
  5. One of the women orders tilefish.
  1. If Lewis orders pork chops, then which one of the following is a complete and accurate list of the entrees any one of which John could order?
  1. roast beef
  2. veal cutlet
  3. roast beef, veal cutlet
  4. roast beef, swordfish
  5. pork chops, roast beef, swordfish
  1. Suppose that the people in each couple both order the same kind of entree as each other rather than order different kinds of entrees. If all other conditions remain the same, and no two women order the same kind of entree, then which one of the following statements could be true?
  1. John orders roast beef.
  2. John orders swordfish.
  3. Kate orders roast beef.
  4. Two of the people order pork chops.
  5. Two of the people order tilefish.

The Action:

We can tell from the first three sentences that the first set is a matching set; we're given 6 people (3 couples—John and Kate, Lewis and Marie, and Nat and Olive), and are asked to match each to one particular entree (pork chops, roast beef, swordfish, tilefish, or veal cutlet). Notice that with 5 entrees and 6 people, there has to be some duplication of entrees. The Key Issues are basic:

1) Who orders what entree?

2) Who can, must, or cannot order the same entree as whom?

The Initial Setup:

To sketch the information, simply write J, K, L, M, N, and O across the top of the page with double lines between K and L, and M and N, in order to visually break the 6 people into their respective couples. If you wish, you can also distinguish between men and women, by writing tone group in capital letters, and the other in lowercase. Add in the list of the entrees off to the side, and you should have something like this to begin with:

The Rules:

Start with the two most concrete rules, Rules 3 and 5.

3) An “S” under Marie will remind us that she opted for the swordfish.

5) “R” under Olive means that it’s roast beef for her.

1) This rule adds a nice touch—it allows the man and woman in each couple to share. One easy way to remember this is to write a “≠” between the members of each couple.

2) You could write “men never order the same,” but you’re always better off being more specific: J≠L≠N.

4) No fish for John or Nat, which means that swordfish and tilefish are out for them. But there are only five entrees to begin with, which means that these two guys are restricted to either pork chops, roast beef, or the veal. Indicate that somewhere in your sketch.

 

Key Deductions: and we have “NO R B” under Nat to indicate that he wants something different than his date, Olive. Our work with combining Rules 2 and 4 will prove quite useful. We should also include Rule 1 in our considerations.

Since Olive, Nat’s better half, orders roast beef (Rule 5), Nat can’t, and Nat’s choices were limited to begin with (Rule 4). Therefore, Nat must order either pork chops or veal cutlet. We can also deduce that Lewis won’t order swordfish since he won’t duplicate Marie’s choice. Notice that while we have definite choices for two out of the three women, we also have lots of information on the men, especially Nat. Once you get the setup to this point, you can bet that more than a couple of answers will spring from John and Nat running out of entrees to order thanks to Rules 1 and 2.

The Final Visualization: Here’s what we have before moving on to the questions:

The Questions:

1. (D)

We deduced that Lewis will not order swordfish since Marie, his date, orders it. However, there’s nothing that prevents him from ordering any of the other entrees, so the other four must be included in the list.

2. (B)

Not much to do but check out each choice. (A) is knocked out by a combination of Rules 3 and 4. (C) violates Rule 2. Rules 1 and 5, taken together, signify that (D) can’t be true, and Rule 1 by itself makes (E) impossible. That leaves (B); Kate would have to order either pork chops or veal cutlet, because that's all Nat can eat, but that’s eminently feasible.

3. (A)

Here’s our Big Deduction: Nat has to order either pork chops or veal cutlet, and choice (A) therefore must be a true statement. (B) and (C) are impossible, while (D) could be true, but need not be (Lewis could order tilefish) and (E) merely could be true as well, but would be false if Kate ordered fish.

4. (E)

John orders the veal, so Nat and Lewis cannot (Rule 2). This makes pork chops Nat’s only choice. Just for the record, Kate ordering pork chops and Lewis ordering roast beef kills choices (A) through (D)

5. (C)

Nat doesn’t order pork chops (no one does), so he orders veal cutlet. Oops, not one of our choices, so we must continue. John now must order the roast beef (now his only choice). Still not an answer choice (though it does rule out choice (A)). Lewis’ choices are now limited to tilefish, whatever that may be. That’s a choice—(C). (B) and (E) could be true, (D) never.

6. (A)

Rule 2 forces Nat to order the veal cutlet, and the same rule mandates that John settle for the roast beef. John’s complete and accurate list is simply roast beef, choice (A).

7. (D)

Deal with rule changes first. Since Marie has swordfish, now so does Lewis. Olive has roast beef; now, so does Nat. Rule 2 is still in effect, so John is left with a choice of pork chops and veal cutlet, which means that Kate is limited to the same. They both can opt for the pork chops, so (D) can be true. (A), (B), (C), and (E) are impossible.

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