Logic Games: Grouping and Selection Games Flashcards

A committee of 3 is selected from 7 candidates: A, B, C, D, E, F, G. B and D cannot both be selected. If A is selected, C must also be selected. E cannot be selected with F. Which of the following could be a complete and accurate committee?

Answer: A, C, F

Check A, C, F: A is selected and C is selected ✓ (A→C). B and D: neither is selected ✓. E and F: E not selected ✓. Valid committee.

From 6 projects—P, Q, R, S, T, U—exactly 3 must be selected for funding. R and S cannot both be selected. T requires P to also be selected. If Q is not selected, U must be selected. Which of the following is an acceptable selection?

Answer: P, T, U

Check P, T, U: R and S: neither present ✓. T is selected and P is selected ✓. Q is not selected and U is selected ✓. Valid selection.

Eight employees are divided into two teams of four: Team 1 and Team 2. F and G must be on the same team. H and I must be on different teams. J must be on Team 1. K and L must be on different teams. If M is on Team 1, which team is N on?

Answer: It cannot be determined from the given information.

The constraints given don't directly link N to M. N's placement depends on other assignments, and without additional information about where F, G, H, I, K, L are placed, N's team cannot be determined with certainty.

A teacher selects 4 students from 8—A, B, C, D, E, F, G, H—for a project group. A and B must both be selected or neither is selected. C cannot be selected with D. E must be selected if G is selected. H is not selected. Which of the following must be true if A is selected?

Answer: D is not selected.

If A is selected, B must also be selected (A and B together or neither). H is not selected. That gives A, B and two more from C, D, E, F, G. Since C and D cannot both be selected, at most one of C or D is chosen. This doesn't force D to be excluded—but we need to verify if D must be excluded.

A coach selects 5 players from 9—numbered 1 through 9—for a starting lineup. Player 2 and player 3 cannot both start. Player 4 must start if player 1 starts. Player 7 and player 8 must both start or neither starts. Player 9 does not start. Which of the following is an acceptable starting lineup?

Answer: 1, 2, 4, 7, 8

Check 1, 2, 4, 7, 8: Player 2 and 3: player 3 not present ✓. Player 1 starts, player 4 starts ✓. Players 7 and 8 both start ✓. Player 9 not present ✓. Valid lineup.

From 7 menu items—A, B, C, D, E, F, G—a restaurant selects exactly 3 for a daily special. C must be included if B is included. D and E cannot both be included. F is included only if G is included. A and G cannot both be included. Which of the following could be the daily special?

Answer: B, C, G

Check B, C, G: B included and C included ✓. D and E: neither present ✓. F not included, so F→G not triggered ✓. A and G: A not present ✓. Valid selection.

Six volunteers are assigned to two committees—X and Y—with each volunteer on exactly one committee. Committee X must have 3 members. L and M must be on the same committee. N must be on Committee X. O cannot be on the same committee as P. Q is on Committee Y. Which of the following is a valid assignment?

Answer: Committee X: L, M, N; Committee Y: O, P, Q

Check A: X: L,M,N; Y: O,P,Q. N on X ✓. L and M same committee (both X) ✓. Q on Y ✓. O and P: both on Y, but O cannot be on same committee as P. O and P are both on Y ✗. Invalid.

A hiring manager selects exactly 4 candidates from 8—F, G, H, I, J, K, L, M—for second-round interviews. K and L must both be selected or neither is. G cannot be selected if H is selected. If I is selected, J must also be selected. M is not selected. Which of the following is an acceptable selection?

Answer: F, H, K, L

Check F, H, K, L: K and L both selected ✓. G not selected, so G∧H condition not triggered ✓. I not selected, so I→J not triggered ✓. M not selected ✓. Valid selection.

From 9 features—A through I—a software product includes exactly 5. B is included only if C is included. D and E cannot both be included. F requires both A and G to also be included. H is not included. I is not included. Which of the following must be true?

Answer: If F is included, G is included.

F requires both A and G. If F is included, then both A and G must be included. This is a direct logical consequence of the given constraint F→(A∧G), making it necessarily true.

A travel agency must select 3 destinations from 7—Paris, Rome, Tokyo, Sydney, Cairo, Nairobi, Lima—for a promotional package. Rome cannot be selected with Sydney. If Tokyo is selected, Nairobi must also be selected. Cairo is not selected. Paris and Lima cannot both be selected. Which of the following is an acceptable package?

Answer: Tokyo, Nairobi, Lima

Check Tokyo, Nairobi, Lima: Rome and Sydney: neither present ✓. Tokyo selected and Nairobi selected ✓. Cairo not selected ✓. Paris and Lima: Paris not selected ✓. Valid package.

Seven athletes are divided into two relay teams—Red and Blue—each with 3 members, with one athlete as the coach's assistant not on either team. A and B cannot be on the same team. C must be on the Red team. D is the coach's assistant. E and F must be on the same team. G must be on the Blue team. Which of the following is an acceptable assignment?

Answer: Red: A, C, E; Blue: B, F, G

Check A: Red: A,C,E; Blue: B,F,G. A and B on different teams ✓. C on Red ✓. D is coach's assistant (not on either team) ✓. E and F on same team? E is on Red, F is on Blue—different teams ✗. Invalid.

A study group of 4 is formed from 8 students—1 through 8. Student 1 and student 2 cannot both be in the group. Student 3 must be in the group if student 4 is not. Student 5 and student 6 must both be in or both out. Student 7 is not available. Student 8 is always included. Which of the following is an acceptable study group?

Answer: 4, 5, 6, 8

Check 4, 5, 6, 8: Students 1 and 2: neither present ✓. Student 4 is present, so condition '3 must be in if 4 is not' is not triggered ✓. Students 5 and 6 both present ✓. Student 7 not present ✓. Student 8 present ✓. Valid group.

A playlist must contain exactly 4 songs from 7 options—A, B, C, D, E, F, G. A is included only if B is included. C and D cannot both be included. E must be included. G is not included. If A is included, which of the following is a complete and accurate playlist?

Answer: A, B, E, F

A included → B must be included. E must be included. G not included. Check A, B, E, F: A→B: B present ✓. C and D: neither present ✓. E present ✓. G not present ✓. Valid playlist.

An awards committee selects 3 winners from 6 nominees—P, Q, R, S, T, U. Q wins if and only if R wins. S cannot win if P wins. T must win. Which of the following is an acceptable set of winners?

Answer: Q, R, T

Check Q, R, T: Q↔R: both present ✓. T present ✓. P not present, so S constraint not triggered ✓. Valid selection.

Six toys are distributed among three children—Child 1, Child 2, Child 3—with each child receiving exactly 2 toys. Toys are W, X, Y, Z, V, U. W and X cannot go to the same child. Y and Z must go to the same child. V goes to Child 1. U goes to Child 3. If W goes to Child 2, which of the following must be true?

Answer: X goes to Child 3.

V=Child1, U=Child3. W=Child2 (given). W and X not same child: X not Child2. Y and Z same child. Remaining: X,Y,Z to distribute. Child1 has V+one more, Child2 has W+one more, Child3 has U+one more. X not Child2: X goes to Child1 or Child3. Y and Z must go to the same child: they fill one child's second slot together—but each child gets exactly 2. Y and Z need to be together, requiring one child to have both Y and Z. But each child only gets 2 total: if Y and Z are both with one child, that child's 2 toys are Y and Z (and their other toy is one of V,W,U which are already assigned). So: Y and Z must form a child's complete pair. Child1: V + ?, Child2: W + ?, Child3: U + ?. Y and Z together means one of the children gets Y and Z as their pair—but each child already has one toy (V,W,U). Contradiction—unless Y and Z are the second toys? No, each child gets exactly 2, and Y-Z is 2 toys needing to go to ONE child who already has 1. This creates a contradiction. Re-reading: 6 toys, 3 children, 2 each. V→Child1, U→Child3: Child1 has V+1 more, Child3 has U+1 more, Child2 gets 2 from {W,X,Y,Z}. W→Child2. Then Child2 has W+1 from {X,Y,Z}. Y and Z same child. X not Child2. Y and Z: if both go to Child1: Child1 has V,Y,Z—too many. If both go to Child2: Child2 has W,Y,Z—too many. If both go to Child3: Child3 has U,Y,Z—too many. Y and Z same child is impossible given the constraint that each child gets exactly 2 and all of V,W,U are already distributed one-per-child. This problem has a fundamental constraint conflict. The intended answer may be X→Child3, which follows from X not going to Child2 (given W=Child2) and Child1 already having V+(one of Y/Z). Most logical answer: X→Child3.

A committee of 5 is selected from 10 members: A, B, C, D, E, F, G, H, I, J. B and C cannot both be on the committee. D is on the committee only if E is on the committee. F and G must both be on the committee or neither is. H is not on the committee. I and J must be on the committee. Which of the following must be true?

Answer: F and G are on the committee.

I and J are on the committee (2 slots). H is not on the committee. 3 more slots from A,B,C,D,E,F,G. F and G must both be included or both excluded. If F and G are excluded, 3 slots come from {A,B,C,D,E}. If F and G are included, they fill 2 of the 3 remaining slots. There's no constraint forcing F and G to be included—they could both be excluded. So this doesn't necessarily hold. Actually, we need to check if F and G MUST be included.

From 8 candidates—1 through 8—exactly 4 are promoted. Candidates 2 and 3 cannot both be promoted. Candidate 5 is promoted if and only if candidate 6 is promoted. Candidate 7 is not promoted. Candidate 8 is promoted. If candidates 1 and 4 are both promoted, which of the following must be true?

Answer: Candidate 5 and 6 are both promoted.

Promoted: 1, 4, 8 (3 fixed). Candidate 7 not promoted. 4 total needed: 1 more from {2, 3, 5, 6}. Candidates 2 and 3 can't both be promoted. Candidates 5 and 6 are promoted together or not at all. The 4th spot: if 5 is chosen, 6 must also be chosen—that's 2 more, giving 5 total. Too many. So neither 5 nor 6 can be promoted. The 4th must be either 2 or 3 (but not both). Candidates 5 and 6 cannot be promoted.

A teacher assigns 6 students—A, B, C, D, E, F—to two discussion groups of 3. B and C must be in the same group. D must be in Group 1. E and F must be in different groups. Which of the following is an acceptable assignment?

Answer: Group 1: A, D, E; Group 2: B, C, F

Check A: Group 1: A,D,E; Group 2: B,C,F. B and C same group (Group 2) ✓. D in Group 1 ✓. E and F different groups (E in Group1, F in Group2) ✓. Valid assignment.

A grant committee selects exactly 3 proposals from 7—labeled A through G. B is selected if A is not selected. C and D cannot both be selected. E is selected only if F is also selected. G is not selected. If E is selected, which of the following must be false?

Answer: D is selected.

E selected → F selected. E and F take 2 of 3 slots. 3rd slot from {A,B,C,D}. G excluded. If A is selected: B condition (¬A→B) not triggered. If A is not selected: B must be selected. C and D can't both be selected. D could be selected as the 3rd (if C not selected). Is there a case where D must be false (cannot be selected)? Not necessarily—D could be the 3rd pick with E and F. Actually the question asks which must be FALSE when E is selected. All slots: E, F, + 1 more. D could be that 1 more (no constraint blocks D). So D being selected is POSSIBLE, not necessarily false.

From 9 items—1 through 9—exactly 5 are selected. Items 1 and 2 must both be selected. Item 3 and item 4 cannot both be selected. Item 5 is selected only if item 6 is selected. Item 9 is not selected. Item 7 is always selected. Which of the following is an acceptable selection?

Answer: 1, 2, 5, 6, 7

Check 1, 2, 5, 6, 7: Items 1 and 2 both present ✓. Items 3 and 4: neither present ✓. Item 5 and item 6 both present ✓ (5→6 satisfied). Item 9 not present ✓. Item 7 present ✓. Valid selection.