14.1 Semantic, Letter & Number Analogies

Analogies: Conceptual Framework & Classification

An analogy is a statement of invariant relational correspondence between two pairs of elements: $A : B :: C : D$, read as "$A$ is to $B$ as $C$ is to $D$". The fundamental rule governing analogies is that the exact logical, mathematical, or semantic rule mapping element $A$ to element $B$ must map element $C$ to element $D$.

+─────────────────────────────────────────────────────────────+
|                    ANALOGY TAXONOMY MATRIX                  |
+─────────────────────────────────────────────────────────────+
| Category           | Governing Principle                    |
+────────────────────+────────────────────────────────────────+
| Semantic / Word    | Conceptual, operational, or linguistic  |
| Letter / Alphabet  | Positional shifts, reverse pairs       |
| Numerical          | Algebraic, exponential, or digital ops |
+────────────────────+────────────────────────────────────────+

1. Semantic / Word Analogies

A semantic analogy connects meanings. Express the first relationship in a short sentence before inspecting the second pair. “A thermometer measures temperature” supplies an instrument-to-measured-quantity relationship. Thermometer : Temperature :: Balance : Mass preserves that order. Balance : Merchant changes the relationship to instrument-to-user and therefore does not complete the same analogy.

Useful relationship types include:

Preserve direction as well as meaning. Page : Book is part-to-whole; Bicycle : Wheel is whole-to-part and reverses the relation. Author : Book concerns creation, while Reader : Book concerns use. Shared nouns do not make those relationships equivalent.

When several relationships look plausible, choose the most specific one that fits both pairs and the supplied context. A key can open a lock, and a password can grant access to an account; the shared relationship is an access mechanism and its target. Do not add technical rules that the prompt does not state. If unfamiliar words appear, use context and the other options rather than guessing from a vaguely related association.

A synonym preserves the relevant sense of a word; an antonym reverses it. “Light” can mean low weight, illumination or a pale colour, so an analogy using it needs enough context to determine the sense. An option that matches a different dictionary meaning can be a distractor. Vocabulary questions and numerical analogies share the same discipline: write the proposed mapping, preserve its order and test it on the entire second pair.

2. Letter / Alphabet Analogies

Alphabet analogies operate on the 26 letters of the English alphabet. Mastery requires immediate mental recall of forward positional ranks, backward positional ranks, and complementary reverse pairs.

The Positional Value Framework

• Forward Positional Rank ($1$ to $26$):
  • Fast Anchor Mnemonics: EJOTY ($E=5, J=10, O=15, T=20, Y=25$) and CFILORUX (multiples of 3: $C=3, F=6, I=9, L=12, O=15, R=18, U=21, X=24$).
• Backward Positional Rank ($26$ to $1$):

$$\text{Backward Rank} = 27 - \text{Forward Rank}$$

The Reverse Complementary Pair Table

Two letters are reverse opposites if their forward alphabetical positions sum strictly to $27$ ($P_1 + P_2 = 27$).

+─────────────────────────────────────────────────────────────+
|            REVERSE COMPLEMENTARY LETTER PAIRS (SUM = 27)    |
+──────┬──────────────┬───────────────┬──────┬────────────────+
| Pair | Forward Ranks| Mnemonic Aid  | Pair | Forward Ranks  |
+──────┼──────────────┼───────────────┼──────┼────────────────+
| A - Z|  1 + 26 = 27 | Azad / Amazon | G - T|  7 + 20 = 27  |
| B - Y|  2 + 25 = 27 | Boy / By      | H - S|  8 + 19 = 27  |
| C - X|  3 + 24 = 27 | Crux / Xerox  | I - R|  9 + 18 = 27  |
| D - W|  4 + 23 = 27 | Dew / Down    | J - Q| 10 + 17 = 27  |
| E - V|  5 + 22 = 27 | Even / Every  | K - P| 11 + 16 = 27  |
| F - U|  6 + 21 = 27 | Full / Fuel   | L - O| 12 + 15 = 27  |
|      |              |               | M - N| 13 + 14 = 27  |
+──────┴──────────────┴───────────────┴──────┴────────────────+

• Letter Shifting Archetypes:
  • Constant shift: $L_i \to L_i + k$ (e.g., $+3$ shift maps $B \to E, K \to N$).
  • Incremental shift: $+1, +2, +3, +4, \dots$
  • Alternating directional shift: $+2, -2, +2, -2, \dots$
  • Reverse pair mapping: $L_i \to (27 - L_i)$ (e.g., $RAIL \to IZRO$).

3. Number Analogies

Number analogies evaluate arithmetic, algebraic, and structural relationships between numerical integers. The core functional archetypes include:

1. Prime Number Sequences: $13 : 17 :: 29 : 31$ (consecutive prime pairs).
2. Squares and Square Variations ($n : n^2 \pm k$):
   • $8 : 65 :: 12 : 145$ (Rule: $n : n^2 + 1$).
   • $9 : 72 :: 11 : 110$ (Rule: $n : n^2 - n$ or $n(n-1)$).
3. Cubes and Cube Variations ($n : n^3 \pm k$):
   • $5 : 124 :: 7 : 342$ (Rule: $n : n^3 - 1$).
   • $4 : 68 :: 6 : 222$ (Rule: $n : n^3 + n$).
4. Product and Sum of Digits:
   • $48 : 32 :: 76 : 42$ (Rule: $4 \times 8 = 32$; $7 \times 6 = 42$).
   • $582 : 15 :: 934 : 16$ (Rule: $5 + 8 + 2 = 15$; $9 + 3 + 4 = 16$).
5. Digital Root Patterns: Summing digits until a single digit remains (e.g., $497 \to 4+9+7 = 20 \to 2+0 = 2$).

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Worked Example: Complementary Letter Analogy

Problem: If the letter group $\text{TRACK}$ is related to $\text{GIZXP}$, determine what the letter group $\text{TRAIN}$ maps to under the identical rule.

• Step 1: Analyze Letter Transitions from $\text{TRACK}$ to $\text{GIZXP}$ (five letters to five letters):
  • $T (20) \to G (7)$: $20 + 7 = 27$ (Reverse opposite pair: $GT$ Road).
  • $R (18) \to I (9)$: $18 + 9 = 27$ (Reverse opposite pair: Indian Railways).
  • $A (1) \to Z (26)$: $1 + 26 = 27$ (Reverse opposite pair: Azad).
  • $C (3) \to X (24)$: $3 + 24 = 27$ (Reverse opposite pair: Crux).
  • $K (11) \to P (16)$: $11 + 16 = 27$ (Reverse opposite pair: Kurukshetra Park).
• Step 2: Formulate Governing Rule:
  • Each letter is mapped directly to its reverse complementary opposite letter (sum of ranks $= 27$).
• Step 3: Apply Rule to $\text{TRAIN}$:
  • $T (20) \to G (7)$
  • $R (18) \to I (9)$
  • $A (1) \to Z (26)$
  • $I (9) \to R (18)$
  • $N (14) \to M (13)$ (Mnemonic: Man)
• Conclusion: $\text{TRAIN}$ is coded as $\text{GIZRM}$.

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