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A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:
Mildred's portrait must be either first or last
There must be exactly one portrait between Nancy and Quentin
Nancy's portrait must come after Lily's but before Quentin's
Which of the following is an acceptable order in which the portraits could be hung?
Any option without Mildred being first or last is immediately taken out. Any option without one space between Nancy and Quentin is taken out, and any option that has Quentin appearing before Nancy is taken out. Any option that doesn't have Lily appearing before Nancy is taken out, leaving only the correct answer as an option.
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A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:
Mildred's portrait must be either first or last
There must be exactly one portrait between Nancy and Quentin
Nancy's portrait must come after Lily's but before Quentin's
If Mildred's portrait is first, which is a complete and accurate list of all the possible portraits that could appear second?
Knowing that Mildred's portrait comes first lets us plot out the possible locations for the Nancy/Quentin block. They can either go in spots three and five or four and six, respectively. (Nancy cannot go in spot two, because Lily has to come before her.) This creates two possibilities for solutions. The first, with Nancy and Quentin in spots three and six gives us Lily in two, and either Peter or Owen in four and the other one in six. The second possibility with Nancy and Quentin in spots four and six gives us the option of Peter, Lily or Owen for spots two and three, and then whomever is left of Peter and Owen for spot five. Therefore, considering both possibilities the only people who could occupy the second spot when Mildred is in the first spot are Owen, Lily and Peter.
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A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:
Mildred's portrait must be either first or last
There must be exactly one portrait between Nancy and Quentin
Nancy's portrait must come after Lily's but before Quentin's
If Nancy's portrait is second, each of the following must be true EXCEPT:
If Nancy is second, we can immediately put Quentin in the fourth spot. Since Lily must come before Nancy, her portrait must be first. With Lily occupying the first spot, Mildred then must be last. We are left with Owen and Peter to fill out the thrid and fifth spots, in either order. Therefore, every answer given must be true EXCEPT that Peter is third - he could be, but he doesn't have to be.
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A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:
Mildred's portrait must be either first or last
There must be exactly one portrait between Nancy and Quentin
Nancy's portrait must come after Lily's but before Quentin's
If Quentin must come before Peter, how many possible orders are there in which the portraits can be hung?
This gives us a new rule, namely that Quentin must come before Peter. We already know that Nancy must appear exactly two spots before Quentin. We also know that Lily has to come somewhere before Nancy. Our complete new rule states that Lily must come somewhere before Nancy, who must appear exactly two spots before Quentin, who must appear somewhere before Peter. When we add all of these up (including the blank spot between Nancy and Quentin) we have already filled out five of the six possible spots. When we take into account that Mildred must be in either the first or last spot, we are left with two options for this super block. Lily, starting off the block, can either go in the first spot and Mildred in the last, or in the second spot with Mildred in the first. In both scenarios Owen fills in the missing spot between Nancy and Quentin. Therefore, there are only two possible solutions that adhere to all of the given rules and the new added rule.
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A gym teacher wants his 6 students to line up in height order, from shortest to tallest. Corrin and Theresa are girls. Ben, Jonathan, Will, and Dan are the boys.
- Will is neither the tallest nor the shortest
- No girl is taller than Jonathan
- Dan is shorter than Corrin, but taller than Theresa
- Ben is the tallest
If Jonathan is the 2nd tallest, what must be true?
If Jonathan is 2nd tallest, that means Will can go anywhere but 6th, 2nd or 1st tallest positions. Since Corrin is taller than Dan, and Theresa is shorter than Dan, each of their positions can be altered by Will. Thus, the only certainty, is that Corrin cannot be the first or second in line, because Theresa and Dan must precede her.
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A gym teacher wants his 6 students to line up in height order, from shortest to tallest. Corrin and Theresa are girls. Ben, Jonathan, Will, and Dan are the boys.
- Will is not the tallest or the shortest.
- No girl is taller than Jonathan
- Dan is shorter than Corrin, but taller than Theresa.
- Ben is the tallest
Which of the following must be true?
This can be answered by process of elimination. Ben being in 6th doesn't effect anyone, as he is always there. If Jonathan is 3rd tallest, the only one who can be the 5th spot is Will - thus, that answer is wrong. Theresa must be the first person in line, so she doesn't affect any positioning. Thus, Corrin could be 3rd, or 4th in line, despite Theresa's position. Finally, if Dan is 2nd in line, Corrin is not guaranteed to be at the 3rd position. Will can be at the third position, pushing Corrin to the 4th.
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A consultant has agreed to see each of his nine clients-- L, M, N, O, P, Q, R, S, T-- once in the next six days, from Monday through Saturday. He arranges his schedule so that he can see at least one of his clients each day, while maintaining the following conditions:
O is always scheduled on a day before R and M.
P is not scheduled for Saturday.
If T is scheduled on a day after O, then S is scheduled on a day after N.
If T is scheduled on a day before O, then R is scheduled on a day before L.
The consultant always sees fewer clients on Friday and Saturday combined than he sees on any other two days of the week combined.
If the consultant sees O on Wednesday, what is the maximum number of clients that he can see after Wednesday?
The correct answer is 4 clients. Placing O on Wednesday does not lead to any particularly unique diagrams. This game is fairly loose-- there are a lot of possible diagrams. The key insight here is the fact that Friday and Saturday can accommodate, at most, two clients total and any other day accommodates at most two.
Here is a possible diagram that maximizes clients after Wednesday:
Mon: T
Tues: Q, P
Wed: S, O
Thurs: R, N
Fri: M
Sat: L
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A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:
Food must always be on either the top or bottom shelves
Art must be directly above or below calculus
Theology is never on the top or the bottom shelves
Calculus and botany must be separated by exactly one category
Which of the following is a possible order of the book categories on the shelves, from top to bottom?
Answers can be quickly eliminated when F is not the first or last letter or when T is the first or last letter. Other answers can be eliminated when A is not directly next to C. Finally, there must be precisely one letter between B and C. Remember, A can be the letter that separates B and C.
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A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:
Food must always be on either the top or bottom shelves
Art must be directly above or below calculus
Theology is never on the top or the bottom shelves
Calculus and botany must be separated by exactly one category
If calculus is assigned the spot directly below the top shelf, then each of the following could be true EXCEPT
Sports could be directly above food if food is on the bottom shelf, but it could never be below food because food must either be on the top shelf or the bottom shelf. No category can be directly below food when it is on the bottom shelf and we know that calculus is directly below the top shelf.
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A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:
Food must always be on either the top or bottom shelves
Art must be directly above or below calculus
Theology is never on the top or the bottom shelves
Calculus and botany must be separated by exactly one category
When calculus is on the bottom shelf, which of the following could be true, but is not required to be true?
When calculus is on the bottom shelf, food must be on the top shelf and botany and art must occupy the fifth and sixth shelves. The remaining categories, however, can be in any order on shelves two, three, and four. Zoology could be on the shelf directly above theology, but it does not have to be.
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A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:
Food must always be on either the top or bottom shelves
Art must be directly above or below calculus
Theology is never on the top or the bottom shelves
Calculus and botany must be separated by exactly one category
If a condition is added that sports must always be on a shelf between botany and calculus, and if all other conditions remain the same, which of the following cannot be true?
When calculus is on the fourth shelf, botany must be on the second or sixth. Since zoology is on the second shelf, botany must be on the sixth. Sports must be on the fifth shelf to be between botany and calculus, and art must be on the second to be adjacent to calculus, which leaves only the top or bottom for theology.
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A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:
Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.
Which one of the following could be an accurate and complete list of the order the shows air along with the newscasters assigned to host them?
The incorrect answers all violate one of the stated conditions:
(P: Adrian, Q: Brett, R: Adrian, S: Brett, T: Calvin) - Adrian's first show must have exactly two shows in between it and Calvin's first show. This has three.
(Q: Adrian, R: Brett, T, Brett, P: Calvin, S: Adrian) - Adrian cannot have any shows after 9:00; the last slot would be 10:00-11:00, a violation of the rule.
(P: Brett, Q: Adrian, S: Adrian, R: Brett, T: Calvin) - R is aired before both S and T; it is aired after S in this case.
(P: Adrian, Q: Adrian, R: Adrian, S: Calvin, T: Calvin) - All of the newcasters must host at least one show, so Brett's absence here is a violation.
The correct answer does not violate any of the stated conditions.
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A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:
Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.
If P is aired from 6:00-7:00 with Brett as the host, which of the following must be true?
Since we are given Brett as the host of the first show, this means that Adrian must host the 7:00-8:00 show and Calvin must host the 10:00-11:00 show in order to put exactly two shows in between their initial shows. Moreover, the shows in between must be hosted by Adrian or Brett because Calvin must have his first performance in the 10:00-11:00 slot. This gives us the following order:
6:00-7:00: P: Brett
7:00-8:00: Adrian
8:00-9:00: Adrian/Brett
9:00-10:00: Adrian/Brett
10:00-11:00: Calvin
If one combines the last two rules, the following order in which the shows air is created:
Q - R - (S/T)
Applying this to the above model, we get:
6:00-7:00: P: Brett
7:00-8:00: Q: Adrian
8:00-9:00: R: Adrian/Brett
9:00-10:00: S/T: Adrian/Brett
10:00-11:00: T/S: Calvin
The correct answer is the only one that must be true. Every way the game is played requires Calvin to host the 10:00-11:00 show. The incorrect answers can happen, but do not necessarily have to in order for the game to work.
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A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:
Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.
Which of the following shows CANNOT be aired in the 8:00-9:00 slot?
Recall from the previous question that combining the last two rules produces the following order in which the shows air:
Q - R - (S/T)
There must be at least three shows that air after Q, so the latest it could possibly air is at 7:00-8:00.
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A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:
Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.
Which one of the following is an accurate and complete list of the shows that Calvin can host?
Because of the rule that two shows must air in between Adrian's first show and Calvin's first show, the only possible slots in which Calvin can host are 9:00-10:00 and 10:00-11:00.
Using the Q - R - (S/T) combination of the last two rules, that means that of these four, only S and T can occupy the last two slots and be hosted by Calvin as a result. Since P has no set rules, there is nothing stopping it from it being put in the last two spots and being hosted by Calvin either.
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A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:
Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.
If Calvin hosts exactly two shows, which of the following must be false?
In order for Calvin to host two shows, Adrian must host the first show in the 6:00-7:00 slot so that two shows can air before Calvin's first show at 9:00-10:00 and allow an opportunity for Calvin to host a second show at 10:00-11:00. That gives us the following model:
6:00-7:00: Adrian
7:00-8:00: Adrian/Brett
8:00-9:00: Adrian/Brett
9:00-10:00: Calvin
10:00-11:00: Calvin
The correct answer must be false in every scenario, which is the case for the assertion that Adrian hosts three shows. Along with Calvin hosting two shows, this would leave no room for Brett to host a show and violate the rule that each of the newcasters must host at least one show.
The incorrect answers all could be true under certain scenarios.
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A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:
Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.
If Q is aired after Brett’s first show, which of the following could be true?
Since the latest spot that Q can air is the 7:00-8:00 slot, it follows that it will air then and Brett will host P, the only show that can air before Q, at 6:00-7:00. In order to satisfy the rule that Adrian's first show must have two shows in between it and Calvin's first show, Adrian must host Q at 7:00-8:00 and Calvin will host the 10:00-11:00 slot. Working off the Q - R - (S/T) order, that puts R in the 8:00-9:00 slot and leaves T and S in the last two spots in any order. That gives us the following model:
6:00-7:00: P: Brett
7:00-8:00: Q: Adrian
8:00-9:00: R: Adrian/Brett
9:00-10:00: S/T: Adrian/Brett
10:00-11:00: T/S: Calvin
Using this model, we can identify all of the incorrect answers as claims that must be false within the context of the question. The correct answer points out correctly that Brett can host both of the shows in the 8:00-9:00 and 9:00-10:00 slots.
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A cruise line company is preparing the itinerary for its upcoming global tour. The cruise will last five months and travel to exactly one of seven countries – C, D, E, F, G, H, I – each month. No country may be visited more than once. The following conditions must hold:
C or D is part of the itinerary, but not both.
If F is part of the itinerary, G is also included.
If E is part of the itinerary, it is visited earlier in the cruise than D and G.
If F is part of the itinerary, it is visited later in the cruise than both E and H.
I is part of the itinerary and must be visited during either the first or fifth month.
Which of the following could be an accurate itinerary, going chronologically from the first month to the fifth?
Each of the incorrect answers violates one of the stated conditions:
(C, G, E, F, I) - E is visited earlier in the cruise than G.
(I, E, D, F, H) - H is visited earlier in the cruise than F.
(C, E, F, D, I) - C and D cannot both be part of the itinerary.
(H, C, E, F, G) - I must be part of the itinerary in either the first or the fifth month.
The correct answer does not violate any of the stated conditions.
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A cruise line company is preparing the itinerary for its upcoming global tour. The cruise will last five months and travel to exactly one of seven countries – C, D, E, F, G, H, I – each month. No country may be visited more than once. The following conditions must hold:
C or D is part of the itinerary, but not both.
If F is part of the itinerary, G is also included.
If E is part of the itinerary, it is visited earlier in the cruise than D and G.
If F is part of the itinerary, it is visited later in the cruise than both E and H.
I is part of the itinerary and must be visited during either the first or fifth month.
Which of the following CANNOT be the country visited in the fifth month of the tour?
There is no possible scenario in which E can be visited in the fifth month. Since D, G, and F can only be included if they are visited after E, that leaves only I, C, and H to fill the itinerary, insufficient for a full itinerary.
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A cruise line company is preparing the itinerary for its upcoming global tour. The cruise will last five months and travel to exactly one of seven countries – C, D, E, F, G, H, I – each month. No country may be visited more than once. The following conditions must hold:
C or D is part of the itinerary, but not both.
If F is part of the itinerary, G is also included.
If E is part of the itinerary, it is visited earlier in the cruise than D and G.
If F is part of the itinerary, it is visited later in the cruise than both E and H.
I is part of the itinerary and must be visited during either the first or fifth month.
If E is the country visited in the fourth month of the tour, which of the following could be true?
Each of the incorrect answers must be false in the above scenario. If E is fourth, that means that there must be three countries visited beforehand and as D, G, F can only be after E, that leaves I, C, and H as the first three countries. I is the first country visited since it can only be visited in the first or fifth month and H or C can go in the second or third months in any order.
This gives us the following model:
I (1) - H/C (2) - C/H (3) - E (4) - (5)
For the fifth month, we can eliminate D since both C and D cannot be part of the itinerary. F can also be eliminated since that would force G to be included as well and there is only one spot after E, where G would have to be placed. That leaves the only possible country that can be visited in the fifth month as G and provides the following model:
I (1) - H/C (2) - C/H (3) - E (4) - G (5)
Looking at the answers, H being visited in the third month is the only one that is possible in this scenario.
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