Missing Number in a 3×3 Matrix

🎯 Learning Objective

Find the missing number in a 3×3 grid by discovering the row or column rule.

💡 Concept

• Test in a FIXED order: (1) row-wise left to right, (2) column-wise top to bottom, (3) diagonal or all-cells
• For each complete row ask: how do the first two numbers make the third? Try +, −, ×, then a² + b
• A rule is real only if it fits EVERY complete row (or column) — one match is a coincidence
• The complete rows are your practice; apply the confirmed rule to the incomplete one
• If rows give nothing, immediately switch to columns — many grids work top-to-bottom
• Let the size of the answer guide you: close values → +/−, much larger → × or squares

🧮 Key Formulas

Common row rules: c = a + b, c = a − b, c = a × b, c = a² + b²

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Test order: rows → columns → diagonal

✏️ Easy Example

Q. Rows of a 3×3 grid: (2, 4, 6), (3, 6, 9), (4, 8, ?). Find the missing number.

1. Each row is a table: n, 2n, 3n
2. Row 1 uses 2 → 2, 4, 6; Row 2 uses 3 → 3, 6, 9
3. Row 3 uses 4 → 4, 8, and 3×4 = 12

Answer: 12

🇮🇳 Real-Life Example

Reading a railway timetable, you spot the pattern 'every train is 15 minutes after the last'. Missing-number grids reward the same habit — find the step, then fill the gap.

📝 Exam-Level Example

Q. Rows: (5, 3, 34), (6, 2, 40), (7, 4, ?). Find the missing number.

1. Plain sum fails (5+3 ≠ 34), so try squares
2. a² + b²: 5² + 3² = 25 + 9 = 34 ✓, 6² + 2² = 36 + 4 = 40 ✓
3. Apply: 7² + 4² = 49 + 16

Answer: 65

📝 Exam-Level Example

Q. Rows: (12, 7, 5), (15, 9, 6), (20, 11, ?). Find the missing number.

1. Rows do not work as first two → third; check the columns instead
2. Column rule: first number = second + third (7+5 = 12 ✓, 9+6 = 15 ✓)
3. So third = first − second = 20 − 11

Answer: 9

🪄 Memory Trick

If rows fail, don't panic — flip to columns before trying exotic rules. And plug the given options into your suspected rule; the one fitting both complete lines is the answer.

⚠️ Common Mistakes

• ❌ Locking a rule after checking only ONE row — always confirm on the second complete row
• ❌ Testing only rows and never trying columns or diagonals
• ❌ Trying random operations in panic instead of the fixed +, −, ×, square order

🏆 Exam Tips

• ✅ Write your candidate rule and test it out loud on both complete rows before applying
• ✅ Answer much bigger than the inputs? Jump straight to × or squares and save time

📌 Summary

• Test order: rows → columns → diagonal
• A rule must fit every complete row/column, not just one
• Try +, −, ×, then a² + b²
• Big answer → ×/squares; close values → +/−