Miguel's online banking password is nine characters long and includes at least one character of each of five types: digits,...

Solution: Password Character Type Analysis

Creating Visual Representation

Password Structure (9 positions):

Position: 1  2  3  4  5  6  7  8  9
Known:    M  ?  G  _  _  _  _  _  _
Type:     U  P  U  ?  ?  ?  ?  ?  ?

Types: D (Digit), P (Punctuation), U (Uppercase), L (Lowercase), O (Other)

Drawing Immediate Inferences

Key Constraints:

1. 'Other' character placement: Exactly one, must be at position 1 or 9
   • Position 1 is M (Uppercase), so Other must be at position 9
2. No consecutive same types: Creates alternating pattern requirement
3. Frequency constraints:
   • Lowercase > any other single type
   • Punctuation < Digits, Lowercase, Uppercase

Seeking the Critical Insight

Determining Character Counts:

• We have 2 Uppercase letters already (positions 1 and 3)
• For Lowercase to be most frequent, it must be at least 3
• This limits other types to maximum 2 each
• Since Punctuation < others and we have 1, Punctuation = 1
• Total must be 9: L=3, U=2, D=2, P=1, O=1

Processing the Solution

Mapping Remaining Positions:

After position 3, we need to place:

• 3 Lowercase letters
• 2 Digits
• 0 more Uppercase or Punctuation
• 1 Other (at position 9)

Alternating Pattern Analysis:

Position 3: U (last Uppercase)
Position 4: Must be L or D (cannot repeat U)
Position 5: Must alternate from position 4
Position 6: Continue alternating
Position 7: Continue alternating
Position 8: Must be L or D
Position 9: O (fixed)

Testing Configuration:

With 3 L's and 2 D's to place in positions 4-8:

• Starting with L at position 4: L-D-L-D-L ✓ (uses exactly 3 L's and 2 D's)
• Starting with D at position 4: D-L-D-L-? (would need 3rd D, but only have 2)

Therefore, positions 4-8 must be: L-D-L-D-L

Final Solution Synthesis

Complete Password Pattern:

Position: 1  2  3  4  5  6  7  8  9
Type:     U  P  U  L  D  L  D  L  O

Answer:

• Fifth character must be: Digit
• Seventh character must be: Digit

Key Insights:

1. The 'Other' character constraint immediately fixes position 9
2. Character count requirements force a unique distribution
3. Alternating constraint with fixed counts determines exact sequence
4. Only one valid configuration exists given all constraints

Exam Strategy:

When facing TPA questions with alternating patterns and count constraints, first determine the fixed positions and character distributions. This often reveals a unique solution without testing multiple scenarios.