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Box/Stack Puzzles - Beginner-Intermediate Level: stack puzzles BEGINNER-INTERMEDIATE

Comprehensive race against clock worksheet covering 20 beginner-intermediate-level box/stack puzzles problems. Worksheet 8 of 30 emphasizes stack puzzles. Master box stacking, stack arrangement, container puzzles through detailed explanations. Difficulty: building on fundamentals with moderate challenges. Tailored for developing preparation.

📝 Worksheet 8 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Beginner-intermediate level

What you'll learn in this worksheet:
Your progress through Box/Stack Puzzles
Worksheet 8 of 30 (26% complete)

Question 1

Five boxes P, Q, R, S, and T of different sizes are stacked vertically (positions 1 to 5 from bottom to top). The stacking rule is: larger boxes must be placed below smaller boxes. Size relationships: - Box P is larger than box Q but smaller than box R - Box S is the smallest - Box T is larger than box R - Box Q is larger than box S If box S is removed from the stack, which box will be at position 4 (counting from the bottom)?
Step-by-step Solution:

1. Establish size relationships:
- P > Q and R > P → So R > P > Q
- S is smallest → S is less than all others
- T > R
- Q > S

2. Combine all inequalities:
- From T > R and R > P > Q, we get: T > R > P > Q
- From Q > S, we get: T > R > P > Q > S

3. Complete size order (largest to smallest):
T > R > P > Q > S

4. Apply stacking rule (larger below smaller):
- Position 1 (bottom): T (largest)
- Position 2: R
- Position 3: P
- Position 4: Q
- Position 5 (top): S (smallest)

5. Remove box S from position 5:

6. Remaining stack (4 boxes):
- Position 1: T
- Position 2: R
- Position 3: P
- Position 4: Q

7. Answer: Box Q is at position 4

Verification:
- T > R ✓ (pos1 vs pos2)
- R > P ✓ (pos2 vs pos3)
- P > Q ✓ (pos3 vs pos4)
- Q > S ✓ (pos4 vs pos5 before removal)
size-comparison, removal, ordering

Question 2

Six boxes A, B, C, D, E, and F are stacked vertically (positions 1 to 6 from bottom to top). The box at stack position n is then placed at seat position n in a circular arrangement of 6 chairs (seats 1 to 6 in clockwise order). Conditions: - Box A is at the bottom of the stack (position 1) - Box B is at the top of the stack (position 6) - Box C is immediately above box D in the stack - In the circular arrangement, box E is opposite box F - Box C is at seat 4 in the circular arrangement Which box sits opposite box A in the circular arrangement?
Step-by-step Solution:

1. Stack to Circle mapping:
- Seat n = Box at stack position n

2. Fixed assignments:
- Stack pos1 = A → Seat1 = A
- Stack pos6 = B → Seat6 = B
- Box C at seat4 (given) → Stack pos4 = C

3. Box C immediately above Box D in stack:
- C at pos4 → D must be at pos3
- Therefore: Stack pos3 = D

4. Remaining positions in stack: pos2 and pos5 for boxes E and F
- Stack pos2 → Seat2
- Stack pos5 → Seat5

5. In circular arrangement, E opposite F:
- In a 6-seat circle, opposite pairs are: (1,4), (2,5), (3,6)
- Seat1 = A, Seat4 = C, Seat3 = D, Seat6 = B
- For E and F to be opposite, they must occupy seats (2,5)
- Therefore: Seat2 and Seat5 = E and F (in some order)

6. Determine which is which:
- Seat2 = Stack pos2 = either E or F
- Seat5 = Stack pos5 = the other one
- Both arrangements are valid

7. Complete the arrangement:
- Stack: pos1=A, pos2=E/F, pos3=D, pos4=C, pos5=F/E, pos6=B
- Circle: Seat1=A, Seat2=E/F, Seat3=D, Seat4=C, Seat5=F/E, Seat6=B

8. Find box opposite A:
- A is at Seat1
- Opposite seat to Seat1 is Seat4 (since 1+3=4)
- Seat4 = C

9. Answer: Box C sits opposite Box A

Final verification:
- A at bottom ✓
- B at top ✓
- C(pos4) immediately above D(pos3) ✓
- E and F at seats 2 and 5 (opposite) ✓
- C at seat4 ✓

Answer: Box C
hybrid, circular_seating, two_step_logic, cat_advanced

Question 3

Eight positions (1 to 8, bottom to top) form a vertical stack. Only five boxes A, B, C, D, and E are placed; the remaining three positions are empty. Conditions: - Position 1 and position 8 are empty - Box A is at position 5 - Box B is three positions below box C (i.e., position of C = position of B + 3) - Box D is immediately above an empty position - Box E is at an even-numbered position Based on these conditions, at which position is box C located?
Step-by-step Solution:

1. Start with fixed information:
- Position 1 = Empty
- Position 8 = Empty
- Position 5 = A

2. List remaining positions for boxes B, C, D, E:
Available: 2, 3, 4, 6, 7

3. Apply "Box B is three positions below Box C":
- Possible (B, C) pairs: (2,5), (3,6), (4,7), (5,8)
- Position 5 is A, position 8 is Empty → (2,5) and (5,8) invalid
- Remaining: (3,6) or (4,7)

4. Apply "Box E is at an even position":
- Even positions available: 2, 4, 6
- Position 6 is a candidate for C in option (3,6) → then E would need another even

5. Apply "Box D is immediately above an empty position":
- Empty positions: 1 and 8 (fixed), plus possibly others
- Positions above empty:
* Above pos1 (empty) → position 2
* Above pos8 (empty) → none (pos9 doesn't exist)
- So D MUST be at position 2 (above empty pos1)

6. Place D at position 2

7. Re-evaluate B and C with D placed:
- Available now: 3, 4, 6, 7
- (B,C) options: (3,6) or (4,7)

8. Place E at even position:
- Even positions available: 4, 6 (2 is taken by D)
- Try (B,C) = (4,7):
* Then B=4, C=7
* E must be at even → E=6
* Remaining position 3 becomes Empty
- Check D at 2 (above empty pos1) ✓
- Check E at 6 (even) ✓

9. Final arrangement (bottom to top):
- Position 1: Empty
- Position 2: D
- Position 3: Empty
- Position 4: B
- Position 5: A
- Position 6: E
- Position 7: C
- Position 8: Empty

10. Answer: Box C is at position 7

Verification: All conditions satisfied ✓
empty-positions, non-adjacent, complex

Question 4

Eight boxes A, B, C, D, E, F, G, and H are arranged in two vertical stacks (Stack 1 and Stack 2), with 4 boxes in each stack. Positions are numbered 1 to 4 from bottom to top in each stack. - Box A is at the top of Stack 1 - Box B is in Stack 2 at position 2 - Box C is immediately below box A in Stack 1 - Box D is at the same position in Stack 2 as box C in Stack 1 - Box E is at the bottom of Stack 1 - Box F is immediately above box D in Stack 2 Which box is at the bottom of Stack 2?
Step-by-step Solution:

1. Stack 1 Analysis:
- Box A at top → Stack1[4] = A
- Box C immediately below A → Stack1[3] = C
- Box E at bottom → Stack1[1] = E
- Remaining box G goes to Stack1[2]

Stack 1 (bottom to top): E, G, C, A

2. Stack 2 Analysis:
- Box B at position 2 → Stack2[2] = B
- Box D at same position as C in Stack1 (position 3) → Stack2[3] = D
- Box F immediately above D → Stack2[4] = F
- Remaining box H goes to Stack2[1] (bottom)

Stack 2 (bottom to top): H, B, D, F

3. Answer: Box H is at the bottom of Stack 2

Verification: All constraints satisfied ✓
multiple-stacks, inter-stack, relative-position

Question 5

Five boxes P, Q, R, S, and T are in a stack numbered 1 to 5 from TOP to BOTTOM (1 is topmost, 5 is bottommost). Given conditions: - Box P is at the top (position 1) - Box Q is at the bottom (position 5) - Box S is immediately above box T - Box R is at position 4 Which box is at position 4 from the top?
Step-by-step Solution:

1. Understanding the numbering system:
- Position 1 = TOP
- Position 5 = BOTTOM

2. Fixed positions:
- Position 1 = P
- Position 5 = Q
- Position 4 = R

3. Remaining positions: 2 and 3 for boxes S and T

4. "S is immediately above T":
- They must occupy consecutive positions with S above T
- Therefore: S at position 2, T at position 3

5. Final arrangement (top to bottom):
- Position 1: P
- Position 2: S
- Position 3: T
- Position 4: R
- Position 5: Q

6. Answer: Box R is at position 4

Verification: All conditions satisfied ✓
numbering-system, top-to-bottom, basic

Question 6

Initially, five boxes A, B, C, D, and E are in Stack 1 (positions 1-5, bottom to top): A at position 1, B at 2, C at 3, D at 4, E at 5. A series of operations is performed: Operation 1: Move the top box of Stack 1 to a new Stack 2 Operation 2: Move the box at position 2 of Stack 1 to top of Stack 2 Operation 3: Move the top box of Stack 1 to top of Stack 2 After these operations, which box is at the bottom of Stack 2?
Step-by-step Solution:
1. Initial State:
- Stack 1 (bottom to top): A(1), B(2), C(3), D(4), E(5)
- Stack 2: Empty

2. Operation 1: Move top box of Stack 1 to Stack 2
- Remove E from Stack 1 (was at position 5)
- Stack 1 after: A(1), B(2), C(3), D(4)
- Stack 2 after: E(1)

3. Operation 2: Move box at position 2 of Stack 1 to top of Stack 2
- Box at position 2 is B
- Remove B from Stack 1
- Stack 1 after: A(1), C(2), D(3) [positions renumber]
- Stack 2 after: E(1), B(2) [B placed on top]

4. Operation 3: Move top box of Stack 1 to top of Stack 2
- Top box of Stack 1 is D (position 3)
- Remove D from Stack 1
- Stack 1 after: A(1), C(2)
- Stack 2 after: E(1), B(2), D(3) [D placed on top]

5. Final Stack 2 (bottom to top): E, B, D

6. Answer: Box E is at the bottom of Stack 2
temporal, operations, dynamic-stacking

Question 7

Five boxes K, L, M, N, and O are stacked vertically (positions 1 to 5 from bottom to top). Each box has a unique weight from the set: 15kg, 25kg, 35kg, 45kg, and 55kg. Stacking rule: Heavier boxes must be placed below lighter boxes. Conditions: - Box O is at the bottom (position 1) - Box K is at the top (position 5) - Box N's weight is greater than Box M's weight, but less than 50kg - The box immediately above Box M has weight less than 30kg - Box L's weight is exactly 25kg What is the weight of Box N?
Step-by-step Solution:

1. Apply stacking rule: Heavier below lighter
- So weights must decrease from bottom to top
- Position 1 (bottom) = heaviest
- Position 5 (top) = lightest

2. Fixed positions:
- Position 1 = Box O (given)
- Position 5 = Box K (given)
- Therefore: O is heaviest (55kg), K is lightest (15kg)

3. Box L's weight is exactly 25kg
- So L must be at position 4 (since 25kg is second lightest)
- Position 4 = L (25kg)

4. Remaining boxes: M and N for positions 2 and 3
- Remaining weights: 35kg and 45kg

5. Constraint: N > M (in weight)
- So N must be heavier than M
- Therefore: N at position 2 (45kg), M at position 3 (35kg)

6. Verify additional constraints:
- "Box N's weight is less than 50kg" → 45kg < 50kg ✓
- "Box immediately above M has weight < 30kg"
- M at position 3 → box above is position 4 = L (25kg) < 30kg ✓

7. Final stack (bottom to top):
- Position 1: O (55kg)
- Position 2: N (45kg)
- Position 3: M (35kg)
- Position 4: L (25kg)
- Position 5: K (15kg)

8. Answer: Box N weighs 45kg

Verification: All conditions satisfied ✓
inequality, weight_range, numerical_deduction, gmat_quant_logic

Question 8

Seven boxes A, B, C, D, E, F, and G are stacked vertically (positions 1 to 7 from bottom to top). Given conditions: - Box C is at position 3 - The product of the positions of boxes C and D is 12 - Box E is at a prime-numbered position - Box F is at a position that is 5 more than twice the position of box G - The sum of the positions of boxes A and B is 8, with box A below box B - Box D is at an even-numbered position - Box G is at an odd-numbered position What is the position of box A?
Step-by-step Solution:

1. Start with fixed information:
- Position 3 = C (given)

2. Product condition: pos(C) × pos(D) = 12
- 3 × pos(D) = 12
- pos(D) = 4

3. Box D is at even position: 4 is even ✓

4. Prime position for Box E: Prime positions are 2, 3, 5, 7
- Position 3 is C, so E can be at 2, 5, or 7

5. F and G relationship: pos(F) = 5 + 2 × pos(G)
- Possible (G, F) pairs: (1,7), (2,9-invalid), (3,11-invalid)
- Therefore: G at 1, F at 7

6. Sum condition: pos(A) + pos(B) = 8, with pos(A) < pos(B) (A below B)
- Possible pairs: (1,7), (2,6), (3,5)
- Position 1 is G, position 3 is C, position 7 is F
- Therefore: (1,7) and (3,5) invalid
- Only possibility: A at 2, B at 6

7. Box E must be at prime position:
- Remaining position after placing A(2), B(6), C(3), D(4), F(7), G(1)
- Only position left is 5 for E
- 5 is prime ✓

8. Final arrangement (bottom to top):
- Position 1: G
- Position 2: A
- Position 3: C
- Position 4: D
- Position 5: E
- Position 6: B
- Position 7: F

9. Answer: Box A is at position 2

Verification of all conditions:
- C at position 3 ✓
- C(3) × D(4) = 12 ✓
- E at position 5 (prime) ✓
- F(7) = 5 + 2×G(1) ✓
- A(2) + B(6) = 8, A below B (2 < 6) ✓
- D at even position (4) ✓
- G at odd position (1) ✓

Answer: Position 2
mathematical, arithmetic, prime-numbers

Question 9

Five boxes A, B, C, D, and E are stacked one above another (bottom to top numbered 1 to 5). Box C is at position 3. Box A is immediately above box B. Box D is at the bottom. Which box is at the top of the stack?
Step-by-step Solution:

1. Create position table (1 = bottom, 5 = top)

2. Direct assignments:
- Box D is at bottom → Position 1 = D
- Box C is at position 3 → Position 3 = C

3. Key constraint: Box A is immediately above Box B
- They must occupy consecutive positions: (1,2), (2,3), (3,4), or (4,5)
- Position 1 is D, Position 3 is C → only (4,5) is available
- Therefore: Position 4 = B, Position 5 = A

4. Remaining box: E goes to Position 2

5. Final stack (bottom to top):
- Position 1: D
- Position 2: E
- Position 3: C
- Position 4: B
- Position 5: A

6. Answer: Box A is at the top

Verification: All constraints satisfied ✓
vertical-stack, immediate-neighbor, basic

Question 10

Seven boxes A, B, C, D, E, F, and G are stacked vertically (positions 1 to 7 from bottom to top). Given conditions: - Box C is at position 3 - The product of the positions of boxes C and D is 12 - Box E is at a prime-numbered position - Box F is at a position that is 5 more than twice the position of box G - The sum of the positions of boxes A and B is 8, with box A below box B - Box D is at an even-numbered position - Box G is at an odd-numbered position What is the position of box A?
Step-by-step Solution:

1. Start with fixed information:
- Position 3 = C (given)

2. Product condition: pos(C) × pos(D) = 12
- 3 × pos(D) = 12
- pos(D) = 4

3. Box D is at even position: 4 is even ✓

4. Prime position for Box E: Prime positions are 2, 3, 5, 7
- Position 3 is C, so E can be at 2, 5, or 7

5. F and G relationship: pos(F) = 5 + 2 × pos(G)
- Possible (G, F) pairs: (1,7), (2,9-invalid), (3,11-invalid)
- Therefore: G at 1, F at 7

6. Sum condition: pos(A) + pos(B) = 8, with pos(A) < pos(B) (A below B)
- Possible pairs: (1,7), (2,6), (3,5)
- Position 1 is G, position 3 is C, position 7 is F
- Therefore: (1,7) and (3,5) invalid
- Only possibility: A at 2, B at 6

7. Box E must be at prime position:
- Remaining position after placing A(2), B(6), C(3), D(4), F(7), G(1)
- Only position left is 5 for E
- 5 is prime ✓

8. Final arrangement (bottom to top):
- Position 1: G
- Position 2: A
- Position 3: C
- Position 4: D
- Position 5: E
- Position 6: B
- Position 7: F

9. Answer: Box A is at position 2

Verification of all conditions:
- C at position 3 ✓
- C(3) × D(4) = 12 ✓
- E at position 5 (prime) ✓
- F(7) = 5 + 2×G(1) ✓
- A(2) + B(6) = 8, A below B (2 < 6) ✓
- D at even position (4) ✓
- G at odd position (1) ✓

Answer: Position 2
mathematical, arithmetic, prime-numbers

Question 11

Five boxes P, Q, R, S, and T are in a stack numbered 1 to 5 from TOP to BOTTOM (1 is topmost, 5 is bottommost). Given conditions: - Box P is at the top (position 1) - Box Q is at the bottom (position 5) - Box S is immediately above box T - Box R is at position 4 Which box is at position 4 from the top?
Step-by-step Solution:

1. Understanding the numbering system:
- Position 1 = TOP
- Position 5 = BOTTOM

2. Fixed positions:
- Position 1 = P
- Position 5 = Q
- Position 4 = R

3. Remaining positions: 2 and 3 for boxes S and T

4. "S is immediately above T":
- They must occupy consecutive positions with S above T
- Therefore: S at position 2, T at position 3

5. Final arrangement (top to bottom):
- Position 1: P
- Position 2: S
- Position 3: T
- Position 4: R
- Position 5: Q

6. Answer: Box R is at position 4

Verification: All conditions satisfied ✓
numbering-system, top-to-bottom, basic

Question 12

Six boxes A, B, C, D, E, and F are arranged in a rack with 2 rows and 3 columns. Row 1 (top): Positions 1, 2, 3 (left to right) Row 2 (bottom): Positions 4, 5, 6 (left to right) Conditions: - Box A is in Row 1 - Box B is directly below box C - Box D is at position 4 (bottom-left) - Box E is to the immediate right of box A - Box F is in Row 2 but not below box A Which box is at position 5?
Step-by-step Solution:

1. Understand the layout:
- Row 1 (top): positions 1 (Col1), 2 (Col2), 3 (Col3)
- Row 2 (bottom): positions 4 (Col1), 5 (Col2), 6 (Col3)
- "Directly below" means same column, consecutive rows
- "Immediate right" means same row, adjacent columns

2. Direct assignment:
- Position 4 = D (given)

3. Place A and E:
- A in Row 1 (positions 1, 2, or 3)
- E immediately right of A
- Possible pairs: (A=1, E=2) or (A=2, E=3)
- Try A=1, E=2 first

4. Place B and C (B directly below C):
- Possible column pairs: (C at pos1, B at pos4), (C at pos2, B at pos5), (C at pos3, B at pos6)
- pos4 is already D, so (C=1, B=4) invalid
- Try C=3, B=6 (Column 3)

5. Place remaining box F:
- Available position: pos5
- F in Row 2 (pos5) and not below A
- A at pos1 (Col1), F at pos5 (Col2) → not below ✓

6. Final arrangement:
- Position 1: A
- Position 2: E
- Position 3: C
- Position 4: D
- Position 5: F
- Position 6: B

7. Answer: Box F is at position 5

Verification: All conditions satisfied ✓
rows-columns, grid-arrangement, spatial

Question 13

Six boxes A, B, C, D, E, and F are arranged in a rack with 2 rows and 3 columns. Row 1 (top): Positions 1, 2, 3 (left to right) Row 2 (bottom): Positions 4, 5, 6 (left to right) Conditions: - Box A is in Row 1 - Box B is directly below box C - Box D is at position 4 (bottom-left) - Box E is to the immediate right of box A - Box F is in Row 2 but not below box A Which box is at position 5?
Step-by-step Solution:

1. Understand the layout:
- Row 1 (top): positions 1 (Col1), 2 (Col2), 3 (Col3)
- Row 2 (bottom): positions 4 (Col1), 5 (Col2), 6 (Col3)
- "Directly below" means same column, consecutive rows
- "Immediate right" means same row, adjacent columns

2. Direct assignment:
- Position 4 = D (given)

3. Place A and E:
- A in Row 1 (positions 1, 2, or 3)
- E immediately right of A
- Possible pairs: (A=1, E=2) or (A=2, E=3)
- Try A=1, E=2 first

4. Place B and C (B directly below C):
- Possible column pairs: (C at pos1, B at pos4), (C at pos2, B at pos5), (C at pos3, B at pos6)
- pos4 is already D, so (C=1, B=4) invalid
- Try C=3, B=6 (Column 3)

5. Place remaining box F:
- Available position: pos5
- F in Row 2 (pos5) and not below A
- A at pos1 (Col1), F at pos5 (Col2) → not below ✓

6. Final arrangement:
- Position 1: A
- Position 2: E
- Position 3: C
- Position 4: D
- Position 5: F
- Position 6: B

7. Answer: Box F is at position 5

Verification: All conditions satisfied ✓
rows-columns, grid-arrangement, spatial

Question 14

Five boxes P, Q, R, S, and T are in a stack numbered 1 to 5 from TOP to BOTTOM (1 is topmost, 5 is bottommost). Given conditions: - Box P is at the top (position 1) - Box Q is at the bottom (position 5) - Box S is immediately above box T - Box R is at position 4 Which box is at position 4 from the top?
Step-by-step Solution:

1. Understanding the numbering system:
- Position 1 = TOP
- Position 5 = BOTTOM

2. Fixed positions:
- Position 1 = P
- Position 5 = Q
- Position 4 = R

3. Remaining positions: 2 and 3 for boxes S and T

4. "S is immediately above T":
- They must occupy consecutive positions with S above T
- Therefore: S at position 2, T at position 3

5. Final arrangement (top to bottom):
- Position 1: P
- Position 2: S
- Position 3: T
- Position 4: R
- Position 5: Q

6. Answer: Box R is at position 4

Verification: All conditions satisfied ✓
numbering-system, top-to-bottom, basic

Question 15

Six family members - Grandfather (G), Father (F), Uncle (U), Son (S), Nephew (N), and Grandson (GS) - have their boxes arranged in two vertical stacks (Stack A and Stack B), with 3 boxes in each stack. Positions are numbered 1 to 3 from bottom to top in each stack. Family relationships (generations from oldest to youngest): - Generation 1 (oldest): Grandfather (G) - Generation 2: Father (F), Uncle (U) - Generation 3: Son (S), Nephew (N) - Generation 4 (youngest): Grandson (GS) Conditions: - No two boxes from the same generation are in the same stack - The oldest generation member is at the bottom of Stack A - Father's box is in Stack B - Son's box is immediately above his son's (Grandson's) box Which box is at the top of Stack A?
Step-by-step Solution:

1. Understand generations:
- Gen 1 (oldest): G
- Gen 2: F, U
- Gen 3: S, N
- Gen 4 (youngest): GS

2. Apply "oldest at bottom of Stack A":
- Stack A, position 1 = G

3. Apply "Father (F) in Stack B":
- F is in Stack B (position TBD)

4. Apply "Son (S) immediately above Grandson (GS)":
- They must be in same stack with S directly above GS
- Possible positions: (GS at 1, S at 2) or (GS at 2, S at 3)
- They cannot be in Stack A because G is at position 1
- So they must be in Stack B
- Place GS at position 1, S at position 2 in Stack B

5. Stack B so far:
- Position 1: GS (gen4)
- Position 2: S (gen3)
- Position 3: F (gen2) - must go here

6. Remaining boxes for Stack A:
- Used: G (Stack A), F, S, GS (Stack B)
- Remaining: U (gen2) and N (gen3)

7. Fill Stack A:
- Position 1: G (gen1)
- Position 2: N (gen3) - can't put U (gen2) adjacent to same gen? No same gen restriction applies to same stack
- Position 3: U (gen2)

8. Verify no same generation in same stack:
- Stack A: gen1 (G), gen3 (N), gen2 (U) → all different ✓
- Stack B: gen4 (GS), gen3 (S), gen2 (F) → all different ✓

9. Final arrangements:
- Stack A (bottom to top): G, N, U
- Stack B (bottom to top): GS, S, F

10. Answer: Box U (Uncle) is at the top of Stack A

Verification: All conditions satisfied ✓
family-relations, generation, real-world

Question 16

Seven boxes P, Q, R, S, T, U, and V are arranged in a vertical stack (positions 1 to 7 from bottom to top). Box P is three positions above box Q. Box R is immediately below box S. Box T is at the bottom. Box U is somewhere between box R and box P. How many boxes are there between box S and box V?
Step-by-step Solution:

1. Create vertical stack (1=bottom, 7=top)

2. Direct assignments:
- Box T at bottom → Position 1 = T

3. Constraint 1: Box P is 3 positions above Box Q
- Possible pairs: (1,4), (2,5), (3,6), (4,7)
- Position 1 is T, so (1,4) invalid
- Try Q at 4, P at 7

4. Constraint 2: Box R immediately below Box S
- They occupy consecutive positions: (1,2), (2,3), (3,4), (4,5), (5,6), (6,7)
- Position 4 is Q, position 7 is P
- Try R at 2, S at 3

5. Constraint 3: Box U between R and P
- R at 2, P at 7 → positions between: 3,4,5,6
- Position 3 is S, position 4 is Q
- Place U at position 5
- Remaining box V goes to position 6

6. Final arrangement (bottom to top):
- Position 1: T
- Position 2: R
- Position 3: S
- Position 4: Q
- Position 5: U
- Position 6: V
- Position 7: P

7. Count boxes between S (pos3) and V (pos6):
- Positions between: 4 and 5 → Boxes Q and U
- Total: 2 boxes

Answer: 2 boxes

Verification:
- T at bottom ✓
- P at 7, Q at 4 → 3 positions above ✓
- R at 2, S at 3 → immediately below ✓
- U at 5 is between R(2) and P(7) ✓
- All boxes used exactly once ✓
vertical-stack, position-gap, counting

Question 17

Five boxes A, B, C, D, and E are stacked vertically (positions 1-5, bottom to top). Their weights follow certain rules: - The heaviest box is at the bottom - Box A is heavier than box B but lighter than box C - Box D is the lightest - Box E is heavier than A but lighter than C - Box B is heavier than D Which box is at position 3?
Step-by-step Solution:

1. Establish weight relationships:
- A > B and C > A → So C > A > B
- D is lightest → D is smallest
- E > A and C > E → So C > E > A
- B > D

2. Combine all inequalities:
- From C > E > A and A > B, we get: C > E > A > B
- And B > D, so: C > E > A > B > D

3. Complete weight order (heaviest to lightest):
C > E > A > B > D

4. Stacking rule: Heaviest at bottom (position 1), lightest at top (position 5)
- Position 1 (bottom): C (heaviest)
- Position 2: E
- Position 3: A
- Position 4: B
- Position 5 (top): D (lightest)

5. Answer: Box A is at position 3

Verification:
- Heaviest (C) at bottom ✓
- A heavier than B (pos3 vs pos4) ✓
- A lighter than C (pos3 vs pos1) ✓
- D lightest at top ✓
- E heavier than A (pos2 vs pos3) ✓
- E lighter than C (pos2 vs pos1) ✓
- B heavier than D (pos4 vs pos5) ✓
weight-comparison, ordering, vertical-stack

Question 18

Seven positions (1 to 7, bottom to top) form a vertical stack. Only five boxes P, Q, R, S, and T are placed; the remaining two positions are empty. Conditions: - Box Q is at the top (position 7) - Box P is at the bottom (position 1) - Box R and Box S have exactly two entities (boxes or empty slots) between them - One empty slot is at position 2 - Box T is immediately above the second empty slot At which position is box T located?
Step-by-step Solution:

1. Fixed positions:
- Position 1 = P (bottom)
- Position 7 = Q (top)
- Position 2 = Empty (first empty slot - given)

2. Remaining positions to fill: 3, 4, 5, 6
- Need to place: R, S, T, and the second empty slot

3. Box T is immediately above the second empty slot:
- Let the second empty slot be at position X
- Then T must be at position X + 1
- Possible X values: 3, 4, 5 (since X+1 ≤ 7 and X+1 cannot be 7 because Q is there)
- X cannot be 6 because T would need to be at 7, but position 7 is Q

4. Try X = 3 (second empty at 3, T at 4):
- Remaining positions: 5 and 6 for R and S
- R and S at 5 and 6 → they are adjacent (0 entities between them)
- This violates "exactly two entities between R and S" ✗

5. Try X = 4 (second empty at 4, T at 5):
- Remaining positions: 3 and 6 for R and S
- Between positions 3 and 6: we have positions 4 and 5
- Position 4 = Empty, Position 5 = T
- That's exactly 2 entities between R and S ✓
- This works!

6. Try X = 5 (second empty at 5, T at 6):
- Remaining positions: 3 and 4 for R and S
- Between positions 3 and 4: no positions between them (0 entities)
- This violates the constraint ✗

7. Therefore, the only valid arrangement is X = 4:
- Position 3: R
- Position 4: Empty (second empty)
- Position 5: T
- Position 6: S

8. Final arrangement (bottom to top):
- Position 1: P
- Position 2: Empty
- Position 3: R
- Position 4: Empty
- Position 5: T
- Position 6: S
- Position 7: Q

9. Verification of all conditions:
- Q at top (7) ✓
- P at bottom (1) ✓
- R at 3, S at 6 → between them: positions 4 (Empty) and 5 (T) → exactly 2 entities ✓
- Empty slot at position 2 ✓
- T at position 5 is immediately above second empty at position 4 ✓

10. Answer: Box T is at position 5

Final Answer: Position 5
vertical, gaps, empty_slots, banking_clerk

Question 19

Six family members - Grandfather (G), Father (F), Uncle (U), Son (S), Nephew (N), and Grandson (GS) - have their boxes arranged in two vertical stacks (Stack A and Stack B), with 3 boxes in each stack. Positions are numbered 1 to 3 from bottom to top in each stack. Family relationships (generations from oldest to youngest): - Generation 1 (oldest): Grandfather (G) - Generation 2: Father (F), Uncle (U) - Generation 3: Son (S), Nephew (N) - Generation 4 (youngest): Grandson (GS) Conditions: - No two boxes from the same generation are in the same stack - The oldest generation member is at the bottom of Stack A - Father's box is in Stack B - Son's box is immediately above his son's (Grandson's) box Which box is at the top of Stack A?
Step-by-step Solution:

1. Understand generations:
- Gen 1 (oldest): G
- Gen 2: F, U
- Gen 3: S, N
- Gen 4 (youngest): GS

2. Apply "oldest at bottom of Stack A":
- Stack A, position 1 = G

3. Apply "Father (F) in Stack B":
- F is in Stack B (position TBD)

4. Apply "Son (S) immediately above Grandson (GS)":
- They must be in same stack with S directly above GS
- Possible positions: (GS at 1, S at 2) or (GS at 2, S at 3)
- They cannot be in Stack A because G is at position 1
- So they must be in Stack B
- Place GS at position 1, S at position 2 in Stack B

5. Stack B so far:
- Position 1: GS (gen4)
- Position 2: S (gen3)
- Position 3: F (gen2) - must go here

6. Remaining boxes for Stack A:
- Used: G (Stack A), F, S, GS (Stack B)
- Remaining: U (gen2) and N (gen3)

7. Fill Stack A:
- Position 1: G (gen1)
- Position 2: N (gen3) - can't put U (gen2) adjacent to same gen? No same gen restriction applies to same stack
- Position 3: U (gen2)

8. Verify no same generation in same stack:
- Stack A: gen1 (G), gen3 (N), gen2 (U) → all different ✓
- Stack B: gen4 (GS), gen3 (S), gen2 (F) → all different ✓

9. Final arrangements:
- Stack A (bottom to top): G, N, U
- Stack B (bottom to top): GS, S, F

10. Answer: Box U (Uncle) is at the top of Stack A

Verification: All conditions satisfied ✓
family-relations, generation, real-world

Question 20

Seven boxes A, B, C, D, E, F, and G are stacked vertically (positions 1 to 7 from bottom to top). Given conditions: - Box C is at position 3 - The product of the positions of boxes C and D is 12 - Box E is at a prime-numbered position - Box F is at a position that is 5 more than twice the position of box G - The sum of the positions of boxes A and B is 8, with box A below box B - Box D is at an even-numbered position - Box G is at an odd-numbered position What is the position of box A?
Step-by-step Solution:

1. Start with fixed information:
- Position 3 = C (given)

2. Product condition: pos(C) × pos(D) = 12
- 3 × pos(D) = 12
- pos(D) = 4

3. Box D is at even position: 4 is even ✓

4. Prime position for Box E: Prime positions are 2, 3, 5, 7
- Position 3 is C, so E can be at 2, 5, or 7

5. F and G relationship: pos(F) = 5 + 2 × pos(G)
- Possible (G, F) pairs: (1,7), (2,9-invalid), (3,11-invalid)
- Therefore: G at 1, F at 7

6. Sum condition: pos(A) + pos(B) = 8, with pos(A) < pos(B) (A below B)
- Possible pairs: (1,7), (2,6), (3,5)
- Position 1 is G, position 3 is C, position 7 is F
- Therefore: (1,7) and (3,5) invalid
- Only possibility: A at 2, B at 6

7. Box E must be at prime position:
- Remaining position after placing A(2), B(6), C(3), D(4), F(7), G(1)
- Only position left is 5 for E
- 5 is prime ✓

8. Final arrangement (bottom to top):
- Position 1: G
- Position 2: A
- Position 3: C
- Position 4: D
- Position 5: E
- Position 6: B
- Position 7: F

9. Answer: Box A is at position 2

Verification of all conditions:
- C at position 3 ✓
- C(3) × D(4) = 12 ✓
- E at position 5 (prime) ✓
- F(7) = 5 + 2×G(1) ✓
- A(2) + B(6) = 8, A below B (2 < 6) ✓
- D at even position (4) ✓
- G at odd position (1) ✓

Answer: Position 2
mathematical, arithmetic, prime-numbers
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