We've covered this topic in our YouTube class. In this post you'll find the class recording, downloadable PDFs, and practice questions.
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Missing Number and Character Reasoning
1. 4-Step Systematic Framework & Core Patterns
Solving geometric number figures and continuous letter patterns requires identifying the structural relationship connecting outer branches to target values[cite: 10]:
- Step 1: Identify Target Node (लक्ष्य पद पहचानें): Determine whether the unknown value \(?\) resides at the central circle, the bottom stem, or an outer peripheral vertex[cite: 10].
- Step 2: Magnitude Comparison (परिमाण परीक्षण): If the center/stem value is substantially larger than outer values, test squares, cubes, or products (\(L^2 - R^2\) or \(T_1 \times T_2\)); if moderate, evaluate sums, differences, and linear multipliers[cite: 10].
- Step 3: Dual Verification (दोहरा सत्यापन): Always ensure the candidate algebraic operator holds consistently on both Figure 1 and Figure 2 before applying it to compute Figure 3[cite: 10].
- Step 4: Continuous Letter Factorization (अक्षर गुणनखंड नियम): Count total units (letters + blank gaps). Factorize the sum into equal symmetrical recurring blocks (e.g., \(15 = 5 \times 3\), \(16 = 4 \times 4\), \(18 = 3 \times 6\), \(20 = 4 \times 5\))[cite: 10].
2. Primary Geometric Archetypes & Formulas
- Y-Shape Stem Algebra: Difference of branch squares: $$\text{Stem} = L^2 - R^2 = (L+R)(L-R)$$[cite: 10] or weighted square differences such as \((L^2 - R^2) \times 2\)[cite: 10].
- Cross / Star Coordinate Difference: Vertical product minus horizontal product: $$\text{Center} = (\text{Top} \times \text{Bottom}) - (\text{Left} \times \text{Right})$$[cite: 10] or vertical difference combined with horizontal sum: $$\text{Center} = (\text{Top} - \text{Bottom}) \times (\text{Left} + \text{Right})$$[cite: 10].
- 3-Branch Star Ratio: Product of upper branches divided by the lower stem: $$\text{Center} = \frac{\text{Branch}_1 \times \text{Branch}_2}{\text{Branch}_3}$$[cite: 10].
- Circular Recurrence Sectors: Clockwise progressive series with successive prime numbers or increments: $$T_{n+1} = T_n \times 2 + P_n \quad (P_n \in \{2, 3, 5, 7, 11\})$$[cite: 10].
3. Worked Reference Examples
- Y-Shape Squares Difference: Given branches \((12, 8) \rightarrow 12^2 - 8^2 = 144 - 64 = 80\)[cite: 10]; \((16, 7) \rightarrow 16^2 - 7^2 = 256 - 49 = 207\)[cite: 10]. For \((25, 21)\): \(25^2 - 21^2 = 625 - 441 = 184\)[cite: 10].
- Central Cross Operation: Outer digits Top \(= 12\), Bottom \(= 5\), Left \(= 3\), Right \(= 6\): Center \(= (12 - 5) \times (3 + 6) = 7 \times 9 = 63\)[cite: 10]. For Top \(= 11\), Bottom \(= 6\), Left \(= 8\), Right \(= 5\): \((11 - 6) \times (8 + 5) = 5 \times 13 = 65\)[cite: 10].
- Continuous Series Division: Total 20 positions: \(g\_fe\_ig\_e\_f\_ii\_f\_ei\_g\)[cite: 10]. Factoring \(20 = 4 \times 5\) gives repeating motif \(\text{gfeii}\)[cite: 10], yielding missing letters \(\text{ifgie}\)[cite: 10].
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