A dance show will include exactly five pieces: Requiem, Smooth Step, Trampoline, Unleashed, and Waltzy. Requiem will be presented immediately...

OWNING THE DATASET

Visual Representation

Positions: [1] - [2] - [3] - [4] - [5]

Pieces: Requiem (R), Smooth Step (S), Trampoline (T), Unleashed (U), Waltzy (W)

Constraints:

1. \(\mathrm{R} \rightarrow \mathrm{S}\) (R immediately before S, must be consecutive)
2. \(\mathrm{U} \ldots \mathrm{T}\) (U before T with at least one piece between)
3. \(\mathrm{W} \neq \text{position 5}\) (W cannot be final)

Immediate Inferences

• RS forms a fixed block that moves together
• U and T cannot be adjacent
• T cannot appear before U
• W can only occupy positions 1-4

UNDERSTANDING THE QUESTION

Task: Find which piece CANNOT be second and which CANNOT be third

Answer choices: All five pieces are options for both columns

PROCESSING THE SOLUTION

Critical Insight

The key is recognizing how the spacing constraint between U and T interacts with position limitations.

Testing Position 2

Can Trampoline be second?

If T is in position 2:

• U must be before T with at least one piece between
• U cannot be in position 1 (no space for a piece between U and T)
• VIOLATION: No valid position for U

Conclusion: T CANNOT be second

Quick verification of others:

• R in position 2 → S in position 3 ✓
• S in position 2 → R in position 1 ✓
• U in position 2 → T can be 4 or 5 with space ✓
• W in position 2 → No constraints violated ✓

Testing Position 3

Can Waltzy be third?

If W is in position 3:

• Positions 1, 2, 4, 5 must contain R, S, U, T
• RS must be consecutive → either positions (1,2) or (4,5)

Case 1: RS occupy positions 1,2

• U and T must occupy positions 4,5
• Since U before T: \(\mathrm{U}=4, \mathrm{T}=5\)
• VIOLATION: No piece between U and T

Case 2: RS occupy positions 4,5

• U and T must occupy positions 1,2
• Since U before T: \(\mathrm{U}=1, \mathrm{T}=2\)
• VIOLATION: No piece between U and T

Conclusion: W CANNOT be third

Quick verification of others:

• R in position 3 → \(\mathrm{S}=4\), can place \(\mathrm{U}=1, \mathrm{W}=2, \mathrm{T}=5\) ✓
• S in position 3 → \(\mathrm{R}=2\), can place \(\mathrm{U}=1, \mathrm{W}=4, \mathrm{T}=5\) ✓
• T in position 3 → \(\mathrm{U}=1\), can place \(\mathrm{W}=2, \mathrm{RS}=4,5\) ✓
• U in position 3 → can place \(\mathrm{RS}=1,2, \mathrm{W}=4, \mathrm{T}=5\) ✓

FINAL SOLUTION SYNTHESIS

Solution Path Recap

1. Identified that RS forms a moveable block
2. Recognized U-T spacing constraint as the key limiting factor
3. For position 2: T cannot have U placed validly before it
4. For position 3: W forces impossible adjacent placement of U and T

Final Answer

• Cannot be presented second: Trampoline
• Cannot be presented third: Waltzy

Key Insights

• The spacing constraint between U and T is the most restrictive condition
• When a piece occupies a middle position, check if remaining pieces can satisfy all constraints
• Adjacent-piece constraints (like RS) combined with spacing constraints create impossible scenarios

Exam Strategy

• Start with the most constrained pieces (here, U and T)
• Test extreme positions first (positions that limit placement options)
• Use logical elimination rather than testing all permutations