Doctor Yamata works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities—lecturing, operating, treating patients, and conducting research. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule must satisfy the following restrictions:
- She performs operations on exactly three mornings.
- If she operates on Monday, she does not operate on Tuesday.
- She lectures in the afternoon on exactly two consecutive calendar days.
- She treats patients on exactly one morning and exactly three afternoons.
- She conducts research on exactly one morning.
- On Saturday she neither lectures nor performs operations.
- Which one of the following must be a day on which Doctor Yamata lectures?
- On Wednesday Doctor Yamata could be scheduled to
- conduct research in the morning and operate in the afternoon
- lecture in the morning and treat patients in the afternoon
- operate in the morning and lecture in the afternoon
- operate in the morning and conduct research in the afternoon
- treat patients in the morning and treat patients in the afternoon
- Which one of the following statements must be true?
- There is one day on which the doctor treats patients both in the morning and in the afternoon.
- The doctor conducts research on one of the days on which she lectures.
- The doctor conducts research on one of the days on which she treats patients.
- The doctor lectures on one of the days on which she treats patients.
- The doctor lectures on one of the days on which she operates.
- If Doctor Yamata operates on Tuesday, then her schedule for treating patients could be
- Monday morning, Monday afternoon, Friday morning, Friday afternoon
- Monday morning, Friday afternoon, Saturday morning, Saturday afternoon
- Monday afternoon, Wednesday morning, Wednesday afternoon, Saturday afternoon
- Wednesday morning, Wednesday afternoon, Friday afternoon, Saturday afternoon
- Wednesday afternoon, Friday afternoon, Saturday morning, Saturday afternoon
- Which one of the following is a pair of days on both of which Doctor Yamata must treat patients?
- Monday and Tuesday
- Monday and Saturday
- Tuesday and Friday
- Tuesday and Saturday
- Friday and Saturday
Next we come face to face with the hectic world of one Doctor Yamata, who performs various activities during the course of a week. Scheduling events or activities during days of a week, mornings and afternoons, isn’t too unusual—this common everyday scenario is the foundation of a number of sets you’ll encounter in your Kaplan course. One odd feature of this one, though, is the actual days of the week; notice that Thursday is missing (probably golf day). Hopefully you didn’t breeze by this little wrinkle. The action mainly involves distributing four activities over the course of a week, which places the set in the grouping camp. Most schedule sets have a heavy sequencing element, but here there’s really only one true sequencing rule, Rule 3. And the questions don’t really hinge on normal sequencing aspects, such as “before” and “after.” So let’s call this a grouping set of distribution with a slight sequencing element attached. We’re asked to distribute activities over the course of a slightly irregular work week. The main Key Issue to deal with is:
1) Which activities must, can, or cannot be performed on the mornings and afternoons of the five days of the week?
The Initial Setup:
A chart indicating the mornings and afternoons of Yamata’s work days (don’t forget, no Thursday!) is in order here, with the activities listed underneath. There are numerous ways to shorthand the activities: For convenience sake, we’ll use L for lecturing, O for operating, T for treating patients, and C for conducting research. Of course, any convention that works for you is okay.
Note that the intro paragraph specifies that exactly one activity will be performed during each morning and afternoon shift, which eliminates the possibility of placing multiple letters in a single box. Our job, then, is to fill each box (time slot) with exactly one letter (activity) as directed by the rules and questions.
1) Rule 1 narrows things down nicely—three of the five boxes in our top morning row will be filled with O’s. We’re not sure which yet, but it’s a start. “OOO” to the right of the morning row in our sketch is one way to remind ourselves of this.
2) is an if-then rule, common to grouping sets of distribution. If O Mon, no O Tues. Naturally, we want to think through the contrapositive, and get that down on the page as well: If O Tues, no O Mon. It’s possible at this point to combine this with Rule 1, since both deal with operating. But let’s hold off for now and wait until Step 4 of the Kaplan Method; we’ll piece it all together below in the Key Deductions section.
3) Here’s the sequencing rule alluded to above: We need to place L’s in the afternoon row on “exactly two consecutive calendar days.” This begs the question “what are ‘consecutive calendar days’ in this setup?” Here’s where the missing Thursday comes into play. In Yamata’s schedule, consecutive calendar days are Monday and Tuesday, Tuesday and Wednesday, and Friday and Saturday—not Wednesday and Friday. Wednesday and Friday appear next to each other in our sketch, but are not “consecutive calendar days.” Those who missed this distinction were in for a rough set. For now, let’s indicate the consecutive L’s on the page, and move on.
4) Rule 4 tells us that we’ll need to fit exactly one T in the morning row and exactly three T’s in the afternoon row. Another significant fact that we’ll come back to in a jiffy.
5) Conducting research, C, gets into the act, or at least into the top morning row, exactly once. Let’s indicate that near our other morning notes.
6) No L’s or O’s on Saturday, which turns out to be a key piece of information that allows us to blow this set wide open.
Before reading these deductions, stop and go back over the set and the rules and try to piece the puzzle together yourself. If you had no trouble the first time, and feel you got most or all of the deductions, that’s great; you probably breezed through the questions, and you need not repeat this exercise. If you had trouble with this set, Step 4 of the Kaplan Method is the skill you need to work on, and this is a great set on which to practice making deductions. Here’s the objective: It may take a few minutes, but it’s possible to deduce a full five of the ten calendar slots, and to narrow the other five down to a few possible activities. When you’ve gone as far as you can go, check your work against the Key Deductions described below.
And away we go. Let’s begin with the afternoon lectures. We need to place two L’s on consecutive calendar days, and we know two major things: There can be no lecture on Saturday Rule 6), so Friday and Saturday is impossible for the two consecutive L’s; and, as discussed above, Wednesday and Friday are not consecutive calendar days. That means that the two afternoon lectures must be on Monday and Tuesday, or on Tuesday and Wednesday. Either way, when, Tuesday afternoon is devoted to lecturing, and the other afternoon lecture must be on Monday or Wednesday. We also know that we must place exactly three T’s in the afternoon slots Rule 4), which along with the two L’s completes the afternoon roster. Well, what slots are available? We just deduced that one L is on Tuesday afternoon. If the other afternoon L is scheduled for Monday, then treating patients would fall on Wednesday, Friday, and Saturday afternoons. If, however, the other afternoon L is on Wednesday, then treating patients would take place on Monday, Friday, and Saturday afternoons. Either way, the good doctor must treat patients on Friday and Saturday afternoons. The other afternoon T will fall on Monday or Wednesday, whichever is not taken up by the remaining afternoon lecture.
What about the morning time slots? The entities are set; we’re to distribute 3 O’s, 1 T, and 1 C into the five morning slots (Rules 1, 4, and 5). There’s lots of information regarding the operating days, and operating takes up three of the five morning slots, so let’s begin there. No O on Saturday—Rule 6 forbids it. That makes it easier: Saturday morning will be devoted to treating patients or conducting research, and we can focus on distributing 3 O’s into the first four days—Monday, Tuesday, Wednesday, and Friday. And Rule 2 narrows down the possibilities even further: If she operates on Monday, then she doesn’t operate on Tuesday, which means she must operate on Wednesday and Friday. Similarly, if she operates on Tuesday, then she doesn’t operate on Monday (the contrapositive of Rule 2), and must again operate on Wednesday and Friday to satisfy Rule 1. Either way, Yamata must operate on Wednesday and Friday mornings, and the third operating morning will be on Monday or Tuesday. Whichever activity is not on Saturday morning, T or C, will fall on Monday or Tuesday morning, the slot not taken by the final O.
The Final Visualization: Whew! It may seem like a lot of time to spend up front, but as
you’ll see, compiling this great information into one sketch makes for quick, stress-free
questions later on. Here’s the load of deductions we have to work with heading into the
Tuesday afternoon is a definite lecture. Armed with our Key Deductions, Q. 1 is a definite five-second point. (A), (C), (D), (E) As we’ve seen, the second afternoon lecture must be on either Monday or Wednesday, so (A) and (C) are possible only, while (D) and (E) are flat out impossible.
As we deduced, operating is a must for Wednesday morning, which narrows the choices down to (C) and (D). The Wednesday afternoon activity must be either lecturing or treating patients, which confirms (C) as the possible schedule for Wednesday.
Another non-if, but this one’s a little more involved. Still, all we need to do is check the choices against the sketch.
(A) Must Yamata treat patients on both shifts of a single day? She treats patients in the afternoon on Saturday, but could conduct research Saturday morning. Friday has T in the afternoon and O in the morning. And the final afternoon T could be on Wednesday, a day on which she operates in the morning. So (A) need not be true.
(B) Must she conduct research and lecture on the same day? No—the two afternoon L’s could be scheduled for Tuesday and Wednesday, both days on which Yamata can perform operations in the morning.
(C) Must she conduct research and treat patients on the same day? No—Yamata could treat patients on Wednesday, Friday, and Saturday afternoons, while conducting research only on Monday morning.
(D) Must she lecture and treat patients on the same day? No—the one morning T can be placed on Saturday, far away from the two afternoon L’s.
(E) We’re left with (E), which must be correct. And in fact, there’s no way to avoid having at least one day containing a lecture and operation. One lecture is always Tuesday, and the other is Monday or Wednesday. If it’s on Monday, then we have L’s Monday and Tuesday, and exactly one operation must fall on either Monday or Tuesday morning. If, however, the second afternoon lecture is on Wednesday, then that’s all we need to know: Yamata must operate on Wednesday morning, so no matter how we space out the afternoon lectures, there will always be at least one day containing both a lecture and operation.
Here’s our first hypothetical—we’re to suppose that Yamata operates on Tuesday. No problem; that means that the days for operations are set at Tuesday, Wednesday, and Friday, and we’re interested in when she can possibly treat patients. According to Rule 4, she must treat patients on exactly three afternoons; that alone eliminates choices (A) and (B), which include only two afternoons. Furthermore, we know from our work up front that she must treat patients Friday and Saturday afternoons; in fact, that’s the answer to Q. 5, and allows us to axe (C), which leaves out Friday afternoon. (D) falls by the wayside, because operating is scheduled for Wednesday morning. (E) is all that remains, and represents a perfectly valid schedule for treating patients.
Right off the sketch: Treating patients is the activity for Friday and Saturday afternoon, choice (E).
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