On each day of the Monday through Friday workweek, exactly two of three employees—George, Hilda, and Irene—must remain on duty in the office to answer tele- phones during the noon to 1 p.m. lunch break. In the scheduling of telephone duty, the following constraints must be met： Hilda and Irene have telephone duty together on Tuesday. George and Hilda have telephone duty together on Thursday. No one can have telephone duty on more than four of the five days of the week.
Question: 1
Which of the following can be the week's telephone duty schedule?
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For acceptability questions, the fastest way to eliminate wrong answers is to check the most concrete rules first. Here, checking the fixed schedules for Tuesday and Thursday (Rules 1 and 2) can quickly eliminate several options before you need to do the more time-consuming counting for Rule 3.
Step 1: Understanding the Concept:
This is an "acceptability" question in a logic game. We must test each of the provided schedules against the given set of rules to find the one schedule that does not violate any rule. Step 2: Key Rules to Check:
1. Tuesday's pair must be \{Hilda, Irene\}.
2. Thursday's pair must be \{George, Hilda\}.
3. No employee can have more than 4 duty days. Step 3: Detailed Explanation:
Let's check each option against the rules:
\begin{itemize}
\item (A) Fails Rule 1: The pair on Tuesday is \{George, Irene\}, but it must be \{Hilda, Irene\}.
\item (B) Fails Rule 2: The pair on Thursday is \{Hilda, Irene\}, but it must be \{George, Hilda\}.
\item (C) Let's check this schedule: M=\{G,I\}, Tu=\{H,I\}, W=\{G,H\}, Th=\{G,H\}, F=\{G,I\}.
\begin{itemize}
\item Rule 1: Tuesday is \{Hilda, Irene\}. This is correct.
\item Rule 2: Thursday is \{George, Hilda\}. This is correct.
\item Rule 3: Let's count the number of duty days for each person:
\begin{itemize}
\item George: Monday, Wednesday, Thursday, Friday (4 days). This is OK.
\item Hilda: Tuesday, Wednesday, Thursday (3 days). This is OK.
\item Irene: Monday, Tuesday, Friday (3 days). This is OK.
\end{itemize}
Since all rules are met, this is a valid schedule.
\end{itemize}
\item (D) Fails Rule 3: Hilda has duty on Monday, Tuesday, Wednesday, Thursday, and Friday, which is 5 days. This exceeds the maximum of 4 days.
\item (E) The OCR for this option appears inconsistent. Based on the options, it seems to be M=\{H,I\}, Tu=\{H,I\}, W=\{G,H\}, Th=\{G,I\}, F=\{G,I\}. This fails Rule 2.
\end{itemize}
Step 4: Final Answer:
Schedule (C) is the only option provided that satisfies all the given constraints.
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Question: 2
If Hilda has telephone duty for exactly two days of the week, which of the following must be true?
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In logic games, when a new condition is added, use it to fill in as much of your diagram or schedule as possible. Often, the new condition will lead to a single, fully determined scenario.
George and Irene have telephone duty together on Wednesday.
George and Hilda have telephone duty together on Friday.
Hilda and Irene have telephone duty together on Wednesday.
Hilda and Irene have telephone duty together on Friday.
Irene has telephone duty for exactly three days of the week.
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The Correct Option isA
Solution and Explanation
Step 1: Understanding the Concept:
This is a conditional question. We must accept the new condition ("Hilda has telephone duty for exactly two days") as true and combine it with the original rules to deduce what else must be true. Step 2: Detailed Explanation:
1. Apply the new condition.
From the original rules, we know Hilda must have duty on Tuesday and Thursday. The new condition states she has duty on exactly two days. Therefore, Tuesday and Thursday must be her only duty days.
2. Deduce the consequences.
If Hilda's only duty days are Tuesday and Thursday, she must be OFF on Monday, Wednesday, and Friday.
3. Determine the schedule for the other days.
Since exactly two people must be on duty each day, and Hilda is off on Monday, Wednesday, and Friday, the other two employees—George and Irene—must be on duty together on all three of those days.
4. Construct the full schedule.
\begin{itemize}
\item Monday: \{George, Irene\}
\item Tuesday: \{Hilda, Irene\} (from original rules)
\item Wednesday: \{George, Irene\}
\item Thursday: \{George, Hilda\} (from original rules)
\item Friday: \{George, Irene\}
\end{itemize}
5. Check the options.
We now look for a statement that must be true based on this fixed schedule.
\begin{itemize}
\item (A) George and Irene have telephone duty together on Wednesday. This is true according to our deduced schedule.
\item (B) George and Hilda have telephone duty together on Friday. This is false; the pair is \{George, Irene\}.
\item (C) Hilda and Irene have telephone duty together on Wednesday. This is false; Hilda is off.
\item (D) Hilda and Irene have telephone duty together on Friday. This is false; Hilda is off.
\item (E) Irene has telephone duty for exactly three days of the week. This is false; Irene has duty on Monday, Tuesday, Wednesday, and Friday (4 days).
\end{itemize}
Step 3: Final Answer:
Given the condition, the only possible schedule requires George and Irene to be on duty together on Wednesday.
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Question: 3
If Hilda and Irene have telephone duty together on Monday and on Wednesday, which of the following must be true?
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The "maximum" or "minimum" rules (like the 4-day limit) are often the key to making deductions after new information is added. Always check if a person has reached their limit, as this will determine their status on the remaining days.
George and Hilda have telephone duty together on Friday.
George and Irene have telephone duty together on Friday.
George has telephone duty on exactly three of the days of the week.
Hilda has telephone duty on exactly three of the days of the week.
Irene has telephone duty on exactly three of the days of the week.
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The Correct Option isB
Solution and Explanation
Step 1: Understanding the Concept:
This is another conditional question. We add the new information about Monday's and Wednesday's schedules to our initial set of rules and deduce the complete schedule. Step 2: Detailed Explanation:
1. Combine all known information to build the schedule.
\begin{itemize}
\item Monday: \{Hilda, Irene\} (from new condition)
\item Tuesday: \{Hilda, Irene\} (from original rule 1)
\item Wednesday: \{Hilda, Irene\} (from new condition)
\item Thursday: \{George, Hilda\} (from original rule 2)
\item Friday: ?
\end{itemize}
2. Apply the "four-day limit" rule to make further deductions.
Let's count the duty days for each employee so far (Monday through Thursday):
\begin{itemize}
\item George: Thursday (1 day)
\item Hilda: Monday, Tuesday, Wednesday, Thursday (4 days)
\item Irene: Monday, Tuesday, Wednesday (3 days)
\end{itemize}
Hilda has now worked her maximum of four days. Therefore, Hilda must be OFF on Friday.
3. Determine Friday's schedule.
Since Hilda is off on Friday, the other two employees, George and Irene, must be on duty. So, Friday's pair is \{George, Irene\}.
4. Check the options against the now complete schedule.
The final schedule is M=\{H,I\}, Tu=\{H,I\}, W=\{H,I\}, Th=\{G,H\}, F=\{G,I\}.
\begin{itemize}
\item (A) George and Hilda have telephone duty together on Friday. False.
\item (B) George and Irene have telephone duty together on Friday. True. This must be true.
\item (C) George has telephone duty on exactly three of the days. False. He works on Thursday and Friday (2 days).
\item (D) Hilda has telephone duty on exactly three of the days. False. She works 4 days.
\item (E) Irene has telephone duty on exactly three of the days. False. She works on Monday, Tuesday, Wednesday, and Friday (4 days).
\end{itemize}
Step 3: Final Answer:
The only statement that must be true based on the deductions is that George and Irene have duty together on Friday.
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Question: 4
If George and Hilda have telephone duty together on Monday and George and Irene have telephone duty together on Friday, any of the following can be true EXCEPT:
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For "EXCEPT" questions, you are searching for the impossible. Systematically test each option. If you can build a valid scenario that includes the statement in the option, it "can be true." The one for which you can't build a valid scenario is the answer.
George and Hilda have telephone duty together on Wednesday.
George and Irene have telephone duty together on Wednesday.
George has telephone duty for four days of the week.
Irene has telephone duty for four days of the week.
Hilda and Irene have telephone duty together for two days of the week.
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The Correct Option isD
Solution and Explanation
Step 1: Understanding the Concept:
This is a "could be true, EXCEPT" question, which means we are looking for the one statement that must be false. We need to establish the partial schedule based on the new conditions and then determine the possible options for the remaining day(s). Step 2: Detailed Explanation:
1. Establish the partial schedule.
\begin{itemize}
\item Monday: \{George, Hilda\} (new condition)
\item Tuesday: \{Hilda, Irene\} (original rule)
\item Wednesday: ?
\item Thursday: \{George, Hilda\} (original rule)
\item Friday: \{George, Irene\} (new condition)
\end{itemize}
2. Analyze the remaining day.
Only Wednesday's schedule is unknown. The pair for Wednesday must be \{G, H\}, \{G, I\}, or \{H, I\}. Let's check the current duty counts (for M, Tu, Th, F):
\begin{itemize}
\item George: M, Th, F (3 days)
\item Hilda: M, Tu, Th (3 days)
\item Irene: Tu, F (2 days)
\end{itemize}
All employees have room to work on Wednesday without exceeding the 4-day limit.
3. Test each option to see if it's possible.
\begin{itemize}
\item (A) Can \{G, H\} work on Wednesday? Yes. The final counts would be G=4, H=4, I=2. This is a valid schedule. So (A) can be true.
\item (B) Can \{G, I\} work on Wednesday? Yes. The final counts would be G=4, H=3, I=3. This is a valid schedule. So (B) can be true.
\item (C) Can George have 4 duty days? Yes, this happens if the Wednesday pair is \{G, H\} or \{G, I\}. So (C) can be true.
\item (D) Can Irene have 4 duty days? Irene currently has 2 duty days (Tu, F). To reach 4 days, she would need to work two more days. But only one day, Wednesday, is left to be scheduled. Therefore, the maximum number of days Irene can work is 3. It is impossible for her to work 4 days.
\item (E) Can \{H, I\} work together for two days? They work together on Tuesday. Can they also work together on Wednesday? Yes. The final counts would be G=3, H=4, I=3. This is a valid schedule. So (E) can be true.
\end{itemize}
Step 3: Final Answer:
The only statement that cannot be true (must be false) is that Irene has telephone duty for four days of the week.
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Question: 5
If there is one pair of employees who have telephone duty together for three of the five days, which of the following must be true?
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For complex "must be true" questions, be exhaustive. You must identify ALL possible scenarios that fit the condition. The correct answer is the one statement that holds true across every single scenario you've built.
George and Hilda have telephone duty together for three days of the week.
Hilda and Irene have telephone duty together for three days of the week.
One of the three employees has telephone duty for exactly two days of the week.
Exactly one of the workers has telephone duty for exactly three days of the week.
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The Correct Option isD
Solution and Explanation
Step 1: Understanding the Concept:
This is a complex conditional question. We must consider all possible scenarios that fit the condition ("one pair works together for three days") and find a statement that is true in every single one of those scenarios. Step 2: Detailed Explanation:
Let's identify the fixed schedules: Tu=\{H,I\}, Th=\{G,H\}. The remaining days are M, W, F.
The condition states a pair works together 3 times. Let's test the three possible pairs:
\begin{itemize}
\item Scenario 1: The pair is \{G, H\.} They already work on Thursday. They must also work on two of the remaining three days (M, W, F). Let's say M and W.
\begin{itemize}
\item M=\{G,H\}, Tu=\{H,I\}, W=\{G,H\}, Th=\{G,H\}, F=?
\item Counts so far: G=3, H=4, I=1. Since H is at her 4-day limit, she cannot work on Friday.
\item Friday must be \{G, I\}.
\item Final Schedule 1: M=\{G,H\}, Tu=\{H,I\}, W=\{G,H\}, Th=\{G,H\}, F=\{G,I\}.
\item Final Counts: G=4, H=4, I=2.
\end{itemize}
\item Scenario 2: The pair is \{H, I\.} They already work on Tuesday. They must also work on two of the remaining three days (M, W, F). Let's say M and W.
\begin{itemize}
\item M=\{H,I\}, Tu=\{H,I\}, W=\{H,I\}, Th=\{G,H\}, F=?
\item Counts so far: G=1, H=4, I=3. Since H is at her limit, she cannot work on Friday.
\item Friday must be \{G, I\}.
\item Final Schedule 2: M=\{H,I\}, Tu=\{H,I\}, W=\{H,I\}, Th=\{G,H\}, F=\{G,I\}.
\item Final Counts: G=2, H=4, I=4.
\end{itemize}
\item Scenario 3: The pair is \{G, I\.} They must work on all three available days: M, W, and F.
\begin{itemize}
\item Final Schedule 3: M=\{G,I\}, Tu=\{H,I\}, W=\{G,I\}, Th=\{G,H\}, F=\{G,I\}.
\item Final Counts: G=4, H=2, I=4.
\end{itemize}
\end{itemize}
All three scenarios lead to a valid schedule. Now we must find a statement that is true in all three cases.
\begin{itemize}
\item (A) George has duty on Wednesday. True in Scenarios 1 and 3, but false in Scenario 2. Not a "must be true".
\item (B) \{G, H\} work together 3 times. Only true in Scenario 1.
\item (C) \{H, I\} work together 3 times. Only true in Scenario 2.
\item (D) One employee has duty for exactly two days. In Scenario 1, Irene works 2 days. In Scenario 2, George works 2 days. In Scenario 3, Hilda works 2 days. This statement is true in every possible scenario.
\item (E) One employee has 3 duty days. This is false in all three scenarios.
\end{itemize}
Step 3: Final Answer:
In every valid schedule that meets the condition, there is exactly one employee who works for only two days.
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Question: 6
Any of the following can be true EXCEPT:
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For general "EXCEPT" questions, the impossibility often stems from a maximum or minimum constraint. Here, the 4-day limit is the key. When testing a scenario, always check the work counts of the most constrained individuals (in this case, Hilda, who is part of both fixed pairs).
One pair of employees has telephone duty together for exactly one day of the week
One pair of employees has telephone duty together for exactly four days of the week
The pair of employees that has telephone duty together on Monday also has telephone duty together on Wednesday
The pair of employees that has telephone duty together on Tuesday also has telephone duty together on Wednesday
The pair of employees that has telephone duty together on Thursday also has telephone duty together on Friday
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The Correct Option isB
Solution and Explanation
Step 1: Understanding the Concept:
This is a "could be true, EXCEPT" question, which means we must find the one statement that is impossible under the original rules. We will test each option by trying to construct a valid schedule that makes the statement true. The one that is impossible is the correct answer. Step 2: Detailed Explanation:
Let's analyze the possibilities for each option. The fixed pairs are Tu=\{H,I\} and Th=\{G,H\}.
\begin{itemize}
\item (A) Can a pair work together exactly once? Yes. Consider the schedule: M=\{G,I\}, Tu=\{H,I\}, W=\{G,H\}, Th=\{G,H\}, F=\{H,I\}. Here, the pair \{G,I\} works together exactly once. This schedule is valid (G=3, H=4, I=3 days). So, (A) can be true.
\item (B) Can a pair work together exactly four times? Let's test the three pairs:
\begin{itemize}
\item Can \{G,H\} work 4 times? They already work on Thursday. They would need to work on all three remaining days (M, W, F). The schedule would have \{G,H\} on M, W, Th, F. Let's count Hilda's days: she would work M, Tu (with Irene), W, Th, F. That's 5 days, which violates the 4-day limit. So this is impossible.
\item Can \{H,I\} work 4 times? They already work on Tuesday. They would need to work on M, W, F. The schedule would have \{H,I\} on M, Tu, W, F. Let's count Hilda's days: she would work M, Tu, W, Th (with George), F. That's 5 days, which violates the 4-day limit. So this is impossible.
\item Can \{G,I\} work 4 times? There are only 3 available days (M, W, F) for them to work together. So it's impossible for them to work 4 times.
\end{itemize}
Since it is impossible for any pair to work together 4 times, this statement cannot be true.
\item (C) Can the same pair work on Monday and Wednesday? Yes. Let the pair be \{G,I\}. Schedule: M=\{G,I\}, Tu=\{H,I\}, W=\{G,I\}, Th=\{G,H\}, F=\{G,H\}. This is a valid schedule (G=4, H=3, I=3 days). So, (C) can be true.
\item (D) Can the Tuesday pair \{H,I\ also work on Wednesday?} Yes. Schedule: M=\{G,H\}, Tu=\{H,I\}, W=\{H,I\}, Th=\{G,H\}, F=\{G,I\}. This is a valid schedule (G=3, H=4, I=3 days). So, (D) can be true.
\item (E) Can the Thursday pair \{G,H\ also work on Friday?} Yes. Schedule: M=\{G,I\}, Tu=\{H,I\}, W=\{G,I\}, Th=\{G,H\}, F=\{G,H\}. This is a valid schedule (G=4, H=3, I=3 days). So, (E) can be true.
\end{itemize}
Step 3: Final Answer:
The only scenario that is impossible is for a pair of employees to have duty together for exactly four days, as it would always force Hilda to work five days.
