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Box/Stack Puzzles - Intermediate-Advanced Level: box order INTERMEDIATE-ADVANCED

Intensive strategic solving 🎯 drill: 20 intermediate-advanced-level box/stack puzzles questions. Worksheet 20 of 30 hones your box order abilities. Practice stacking logic, box hierarchy, stack puzzles under timed conditions. Best for advanced developing students seeking advanced concepts with increasing complexity.

📝 Worksheet 20 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate-advanced level

What you'll learn in this worksheet:
Your progress through Box/Stack Puzzles
Worksheet 20 of 30 (66% complete)

Question 1

Nine boxes A, B, C, D, E, F, G, H, and I are arranged in three vertical stacks (Stack X, Y, Z) with 3 boxes each. Positions are numbered 1-3 from bottom to top in each stack. Conditions: - Box A is in Stack X at position 2 - Box B is at the same position in Stack Y as box C in Stack Z - Box C is at the top of Stack Z - Box D is in Stack X at the bottom - The sum of positions of box E in Stack Z and box F in Stack X equals 4 - Box G is immediately above box H in Stack Y - Box I is in Stack Z at position 2 In which stack and position is box E located?
Step-by-step Solution:

1. Direct assignments from given conditions:
- Stack X, position 2 = A
- Stack X, position 1 = D (bottom)
- Stack Z, position 3 = C (top)
- Stack Z, position 2 = I

2. From condition 2: B is at same position in Y as C in Z
- C at position 3 in Z → B at position 3 in Y

3. From condition 6: G immediately above H in Stack Y
- They must occupy consecutive positions: (1,2) or (2,3)
- Position 3 is B, so G and H must be at positions 1 and 2
- H at 1 (bottom), G at 2 (immediately above)

4. Stack Y so far:
- Position 1: H
- Position 2: G
- Position 3: B

5. Remaining boxes: E and F for Stacks X and Z
- Stack X has position 3 available
- Stack Z has position 1 available

6. From condition 5: pos(E in Z) + pos(F in X) = 4
- E can only be at Z position 1 (since Z positions 2 and 3 are I and C)
- F can only be at X position 3 (since X positions 1 and 2 are D and A)
- Therefore: 1 + 3 = 4 ✓

7. Final arrangements:
- Stack X: D(1), A(2), F(3)
- Stack Y: H(1), G(2), B(3)
- Stack Z: E(1), I(2), C(3)

8. Answer: Box E is in Stack Z at position 1

Verification: All 7 conditions satisfied ✓
three-stacks, inter-stack, mathematical

Question 2

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 3

Nine boxes A, B, C, D, E, F, G, H, and I are arranged in three vertical stacks (Stack X, Y, Z) with 3 boxes each. Positions are numbered 1-3 from bottom to top in each stack. Conditions: - Box A is in Stack X at position 2 - Box B is at the same position in Stack Y as box C in Stack Z - Box C is at the top of Stack Z - Box D is in Stack X at the bottom - The sum of positions of box E in Stack Z and box F in Stack X equals 4 - Box G is immediately above box H in Stack Y - Box I is in Stack Z at position 2 In which stack and position is box E located?
Step-by-step Solution:

1. Direct assignments from given conditions:
- Stack X, position 2 = A
- Stack X, position 1 = D (bottom)
- Stack Z, position 3 = C (top)
- Stack Z, position 2 = I

2. From condition 2: B is at same position in Y as C in Z
- C at position 3 in Z → B at position 3 in Y

3. From condition 6: G immediately above H in Stack Y
- They must occupy consecutive positions: (1,2) or (2,3)
- Position 3 is B, so G and H must be at positions 1 and 2
- H at 1 (bottom), G at 2 (immediately above)

4. Stack Y so far:
- Position 1: H
- Position 2: G
- Position 3: B

5. Remaining boxes: E and F for Stacks X and Z
- Stack X has position 3 available
- Stack Z has position 1 available

6. From condition 5: pos(E in Z) + pos(F in X) = 4
- E can only be at Z position 1 (since Z positions 2 and 3 are I and C)
- F can only be at X position 3 (since X positions 1 and 2 are D and A)
- Therefore: 1 + 3 = 4 ✓

7. Final arrangements:
- Stack X: D(1), A(2), F(3)
- Stack Y: H(1), G(2), B(3)
- Stack Z: E(1), I(2), C(3)

8. Answer: Box E is in Stack Z at position 1

Verification: All 7 conditions satisfied ✓
three-stacks, inter-stack, mathematical

Question 4

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 5

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 6

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 7

Six boxes A, B, C, D, E, F are stacked vertically (positions 1-6, bottom to top). Each box has a different color: Red, Blue, Green, Yellow, White, and Pink. Given conditions: - Box A is Red and at position 4 - The Blue box is immediately above the Green box - Box B is Yellow and at the bottom (position 1) - Box C is at position 5 - The White box is at position 5 - The Pink box is at the top (position 6) - Box D is Blue What is the color of box C?
Step-by-step Solution:

1. Direct assignments from conditions:
- Position 1 = Box B (Yellow) [bottom]
- Position 4 = Box A (Red)
- Position 5 = Box C (also White box)
- Position 6 = Pink box

2. Blue immediately above Green:
- Possible consecutive pairs: (1,2), (2,3), (3,4), (4,5), (5,6)
- Position 4 is Red, position 5 is White, position 6 is Pink
- Position 1 is Yellow (B)
- Therefore, Green and Blue must be at positions 2 and 3
- Green at 2, Blue at 3 (Blue above Green) ✓

3. Box D is Blue:
- Blue at position 3 → Box D at position 3

4. Remaining box E:
- All positions filled: 1(B), 2(E/Green), 3(D/Blue), 4(A/Red), 5(C/White), 6(F/Pink)
- Therefore, box E must be at position 2 (Green)

5. Final stack (bottom to top):
- Position 1: B (Yellow)
- Position 2: E (Green)
- Position 3: D (Blue)
- Position 4: A (Red)
- Position 5: C (White)
- Position 6: F (Pink)

6. Answer: Box C is White

Verification of all conditions:
- A at position 4, Red ✓
- Blue (pos3) immediately above Green (pos2) ✓
- B at bottom, Yellow ✓
- C at position 5 ✓
- White box at position 5 ✓
- Pink at top ✓
- D is Blue ✓
- All colors unique ✓
- All positions filled ✓
color-coding, multi-parameter, vertical-stack

Question 8

Five books A, B, C, D, and E are arranged horizontally on a shelf from left to right (positions 1 to 5). Given conditions: - Book A is at the leftmost position - Book B is immediately to the right of Book C - Book D is at the rightmost position - Book E is at position 4 Which book is at position 3?
Step-by-step Solution:

1. Fixed positions:
- Position 1 = A (leftmost)
- Position 5 = D (rightmost)
- Position 4 = E

2. Remaining positions: 2 and 3 for books B and C

3. "B is immediately to the right of C":
- They must be consecutive with C on left, B on right
- Therefore: C at position 2, B at position 3

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

5. Answer: Book B is at position 3

Verification: All conditions satisfied ✓
horizontal-arrangement, left-right, basic

Question 9

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 10

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 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

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 13

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 14

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 15

Five books A, B, C, D, and E are arranged horizontally on a shelf from left to right (positions 1 to 5). Given conditions: - Book A is at the leftmost position - Book B is immediately to the right of Book C - Book D is at the rightmost position - Book E is at position 4 Which book is at position 3?
Step-by-step Solution:

1. Fixed positions:
- Position 1 = A (leftmost)
- Position 5 = D (rightmost)
- Position 4 = E

2. Remaining positions: 2 and 3 for books B and C

3. "B is immediately to the right of C":
- They must be consecutive with C on left, B on right
- Therefore: C at position 2, B at position 3

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

5. Answer: Book B is at position 3

Verification: All conditions satisfied ✓
horizontal-arrangement, left-right, basic

Question 16

Six boxes P, Q, R, S, T, and U are in a vertical stack (positions 1-6, bottom to top). Given conditions: - If P is above Q, then R is at an even position - S is exactly in the middle of the stack - U is not adjacent to S - T is at the bottom - P is two positions above Q Based on these conditions, which box can NEVER be at the top (position 6)?
Step-by-step Solution:

1. Fixed positions from given conditions:
- T is at bottom → Position 1 = T
- S is exactly in the middle → For 6 boxes, middle positions are 3 or 4
- P is two positions above Q → P's position = Q's position + 2

2. Determine valid arrangement:
- Since P = Q + 2, possible pairs: (1,3), (2,4), (3,5), (4,6)
- Position 1 is T, so (1,3) invalid
- Try Q=2, P=4:
* Then S can be at position 3 (middle)
* R must be at even position (2,4,6) but 2 and 4 taken → R=6
* U must not be adjacent to S (pos3) → cannot be at 2 or 4 → U=5
* All positions filled: 1=T, 2=Q, 3=S, 4=P, 5=U, 6=R ✓

3. Test if each box can be at top (position 6):
- T: Fixed at bottom → Cannot be at top ❌
- Q: If Q at 6, then P would be at 8 (invalid) → Cannot be at top ❌
- P: Can be at top if Q=4, P=6 (with S=3, R=2, U=5) → Possible ✓
- R: Can be at top as shown in our arrangement → Possible ✓
- S: Can be at top if S=6 (but then middle would be 3, possible with adjustments) → Possible ✓
- U: Can be at top with different arrangement → Possible ✓

4. Only Q and T cannot be at top:
- T is explicitly fixed at bottom
- Q is mathematically impossible at top due to P=Q+2 constraint

5. The question asks "which box" (singular):
- Since T is explicitly stated to be at bottom, it's obvious
- Q is the non-obvious answer that requires deduction
- Therefore, Q is the intended answer

Answer: Box Q can never be at the top

Verification:
- In any valid arrangement, Q's maximum position is 4 (when P=6)
- Q at position 6 would require P at 8 (outside stack)
- Therefore Q is impossible at top ✓
conditional, either-or, complex-logic

Question 17

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 18

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 19

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 20

Nine boxes A, B, C, D, E, F, G, H, and I are arranged in three vertical stacks (Stack X, Y, Z) with 3 boxes each. Positions are numbered 1-3 from bottom to top in each stack. Conditions: - Box A is in Stack X at position 2 - Box B is at the same position in Stack Y as box C in Stack Z - Box C is at the top of Stack Z - Box D is in Stack X at the bottom - The sum of positions of box E in Stack Z and box F in Stack X equals 4 - Box G is immediately above box H in Stack Y - Box I is in Stack Z at position 2 In which stack and position is box E located?
Step-by-step Solution:

1. Direct assignments from given conditions:
- Stack X, position 2 = A
- Stack X, position 1 = D (bottom)
- Stack Z, position 3 = C (top)
- Stack Z, position 2 = I

2. From condition 2: B is at same position in Y as C in Z
- C at position 3 in Z → B at position 3 in Y

3. From condition 6: G immediately above H in Stack Y
- They must occupy consecutive positions: (1,2) or (2,3)
- Position 3 is B, so G and H must be at positions 1 and 2
- H at 1 (bottom), G at 2 (immediately above)

4. Stack Y so far:
- Position 1: H
- Position 2: G
- Position 3: B

5. Remaining boxes: E and F for Stacks X and Z
- Stack X has position 3 available
- Stack Z has position 1 available

6. From condition 5: pos(E in Z) + pos(F in X) = 4
- E can only be at Z position 1 (since Z positions 2 and 3 are I and C)
- F can only be at X position 3 (since X positions 1 and 2 are D and A)
- Therefore: 1 + 3 = 4 ✓

7. Final arrangements:
- Stack X: D(1), A(2), F(3)
- Stack Y: H(1), G(2), B(3)
- Stack Z: E(1), I(2), C(3)

8. Answer: Box E is in Stack Z at position 1

Verification: All 7 conditions satisfied ✓
three-stacks, inter-stack, mathematical
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