🔠 Patterns & Series (Reasoning)

Day 1 | ⏱️ 1.5 Hours | TCS Interview Prep

📌 Subtopics

• Word Patterns: Identifying patterns in words, letter rearrangements
• Letter Series: Next letter in sequence, alphabetical position-based logic
• Number Series: Arithmetic, Geometric, Fibonacci, Prime-based, and mixed series
• Coding-Decoding: Letter shifting, number substitution, symbol-based coding

📖 Comprehensive Notes

1. Word Patterns & Letter Series

• Alphabet Positions: Make sure to write out/recall:
  • Forward positions: $A=1, B=2, \dots, Z=26$.
  • Reverse positions: $A=26, B=25, \dots, Z=1$.
  • Positional relationship: $\text{Forward Position} + \text{Reverse Position} = 27$.
• Letter Series: Find the next term by checking differences between adjacent letters.
  • Example: $B, D, G, K, ?$
  • Difference: $B \xrightarrow{+2} D \xrightarrow{+3} G \xrightarrow{+4} K \xrightarrow{+5} P$.

2. Number Series

Look for these common patterns in order:

1. Prime Numbers: $2, 3, 5, 7, 11, \dots$
2. Difference (Arithmetic Series): If the series grows slowly, find the difference between consecutive terms. If the first differences don’t show a pattern, check the difference of differences (double difference).
3. Multiplication/Division (Geometric Series): If the series grows rapidly, look for multiplication factors.
4. Squares/Cubes: Look for perfect squares/cubes, or shifts like $n^2 \pm 1, n^3 \pm 1, n^2 + n$.
5. Alternating Series: Check odd-indexed terms and even-indexed terms as two separate series.

Example: Find the next number in the series: $4, 11, 30, 67, 128, ?$ Let’s analyze the relation to cubes:

• $1^3 + 3 = 4$
• $2^3 + 3 = 11$
• $3^3 + 3 = 30$
• $4^3 + 3 = 67$
• $5^3 + 3 = 128$
• Next term: $6^3 + 3 = 216 + 3 = 219$.

3. Coding-Decoding

• Letter Shifting: Letters are shifted forward or backward by a constant number or a sequence (e.g., $+1, +2, +3$).
• Opposite/Reverse Coding: Replacing a letter with its reverse paired letter (e.g., $A \rightarrow Z, B \rightarrow Y$).
• Fictitious Language Coding (Deciphering Message Codes):
  • E.g., “ko to pa” means “cats are wild” and “pa lo de” means “wild dogs bark”.
  • Identify common words in sentences and their corresponding common code words (here “pa” is the only common code, and “wild” is the only common word, so “pa” $=$ “wild”).

Example: If in a certain code $\text{MYSTIFY}$ is written as $\text{NZTUJGZ}$, how is $\text{NEMESIS}$ written? Check pattern: $M(+1)\rightarrow N$, $Y(+1)\rightarrow Z$, $S(+1)\rightarrow T$, $T(+1)\rightarrow U$, etc. (Each letter is shifted by $+1$). Apply to $\text{NEMESIS}$: $N \rightarrow O$, $E \rightarrow F$, $M \rightarrow N$, $E \rightarrow F$, $S \rightarrow T$, $I \rightarrow J$, $S \rightarrow T$. Answer: $\text{OFNFTJT}$.

🌐 Learning Resources

✅ Practice Links

📝 Common Series Types

Arithmetic:  2, 5, 8, 11, 14  (common diff = 3)
Geometric:   3, 6, 12, 24, 48 (common ratio = 2)
Fibonacci:   1, 1, 2, 3, 5, 8, 13
Prime:       2, 3, 5, 7, 11, 13
Squares:     1, 4, 9, 16, 25, 36
Cubes:       1, 8, 27, 64, 125
Mixed:       +1, ×2, +3, ×4 ... (alternating operations)

Coding-Decoding Tip: Map letters to numbers (A=1, B=2… Z=26) and look for +/- shifts

🔗 Related Topics

• Series and Progressions — Math AP/GP progressions (distinct but related series concept)
• Odd Man Out and Analogy — Similar classification logic using sequences/patterns
• Strings — Algorithms for manipulating and pattern matching strings
• Arrays and Hashing — Working with sequences, indices, and pattern counts in coding
• Greedy Number Theory and Bit Manipulation — Binary and math patterns