Missing Number
🔒 Log in to trackHow missing-number puzzles work
🔒 Log in to trackA figure — two square grids, or concentric circles, or a triangle of numbers — carries a hidden rule connecting its cells. One cell is replaced by '?'. Your job is to extract the rule from the complete instances and apply it once.
The examiner's design guarantees the rule: the same relationship must hold in every complete row/segment. So the method is mechanical:
- Read two complete rows (or segments) side by side.
- Test simple candidates in order: sum, difference, product, 'multiply then add/subtract a constant', square, digit-sum. CGL rules are almost always one of these.
- Confirm the candidate on the second complete row. A rule that works on one row but not the other is wrong — this single check eliminates nearly every error.
- Apply the confirmed rule to the incomplete row.
Speed note: start from the biggest number in the row and ask 'how do the other two build it?' — most rules build the largest cell from the smaller ones.
Detailed notes
One hidden rule, found by systematic guessing
A figure — grids, circles, triangles — carries a hidden rule connecting its numbers; one cell is missing. You are not looking for THE rule, only for A simple rule that fits every complete row/segment and gives one of the options.
The standard search order. For each complete row (or matching pair of cells), test in this sequence and stop at the first rule that fits everywhere:
- Sums and differences: third = first + second, or their difference.
- Products and quotients: third = first × second (watch a constant ×, like 2ab).
- Squares and near-squares: third = a², a² + b, a² + b², a² − 1 — numbers like 25, 26, 48, 49 are square magnets.
- Mixed: first² ± second, (first + second)², first × second ± constant.
Fit everywhere, not somewhere. A rule that works on one row is a coincidence. Every complete row must reproduce its third number (and every complete grid must reproduce the other grid's numbers) before you trust it on the missing cell. Two rows fitting 'a + b' while the third fits 'a × b' means the rule is wrong — real exam rules are uniform.
Two-grid puzzles. When two complete grids sit beside an incomplete one, the rule lives inside each grid (usually row by row), and the second grid simply repeats the first grid's rule with different numbers. Solve grid 1 completely first, verify on grid 2, then apply to grid 3.
Segments (circles, triangles). Match positions: opposite segments, or corner-vs-centre. Common segment rules: opposite pairs share a constant sum; centre = sum (or product) of corners; each ring adds a constant.
Option-driven pruning. If the missing cell must be, say, odd, every even option dies — check parity first. If rows are increasing, a decreasing option dies. Prune before deep work; with two options left, often only one fits a plausible rule.
Anchor on the biggest number. The largest cell of a row is almost always the output. Try to rebuild it from the smaller cells: 26 from 5 and 1 (5² + 1), 21 from 4 and 5 (4 × 5 + 1). Reading left to right hides the rule; rebuilding the largest cell exposes it.
Size tells you the operation. Compare the output with the inputs: output ≈ sum → addition family; output ≈ product → multiplication family; output ≈ first² → square family; output smaller than both → difference, average or a digit rule. One glance at the size cuts the search list in half.
Why distractors look right. Exam setters plant the value of the most tempting wrong rule in the options: the bare product when the rule is product + 1, the plain square when it is square − 1, the unreversed sum when the rule reverses. If your answer equals a 'simpler' version of the rule, re-check that the simpler rule really fails a complete row.
Quick revision
- Test rules in order: + / − → × / ÷ → squares → mixed (a² ± b, (a+b)²...).
- The rule must fit EVERY complete row/segment — one fit is a coincidence.
- Two grids: solve grid 1, verify on grid 2, apply to grid 3.
- Segments: match opposite positions; centre often = sum/product of corners.
- Prune by parity and trend before hunting the rule.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Third number from two: linear rulesvery common3 practice Q
Rows of two/three numbers where the third is a plain sum, difference or multiple of the first.
- Test c = ka (constant multiple) and c = a ± b.
- k must be the same in every row.
- Apply to the incomplete row.
Why it works: the simplest rules are linear, so they are tried first.
Example: Rows: 4 → 20, 6 → 30, 5 → ?
Second = 5 × first in both rows → 5 × 5 = 25.
Type 2: Square-based rulesvery common3 practice Q
Third numbers near perfect squares (25, 26, 35, 48, 49...).
- Test c = a² ± b, c = a² ± 1, c = a² + b².
- Square the first number and see what the leftover equals.
- Verify on all rows before answering.
Why it works: squares plus a small correction are a favourite exam construction.
Example: Rows: 3 → 10, 4 → 17, 5 → ?
10 = 3² + 1, 17 = 4² + 1 → 5² + 1 = 26.
Type 3: Two-grid analogycommon3 practice Q
Two complete grids and a third with a missing cell.
- Find grid 1's row rule.
- Verify it reproduces grid 2 exactly.
- Apply it to grid 3's incomplete row.
Why it works: the second grid is a free verification of the rule.
Example: Grid 1 rows: (2,3,7), (3,4,13). Grid 2 rows: (4,5,?), (5,6,31).
Grid 1: c = a×b + 1 (7, 13). Grid 2 row 2: 5×6+1 = 31 ✓ → ? = 4×5+1 = 21.
Type 4: Verify-then-answer disciplinecommon3 practice Q
Any figure where several rules 'almost' fit, or a cross/2×2 layout.
- Candidate rule must fit every complete row/segment AND match an option.
- For 2×2 crosses: test diagonal products and sums too.
- Reject a rule that fits two rows but fails the third.
Why it works: exam figures are built from one uniform rule; near-misses are bait.
Example: Rows: 9 → 81, 12 → 144, 15 → ?
Both rows square the first number → 15² = 225.
Formulas
Two agreeing instances pin the rule; one instance pins nothing.
Test sums/products first — they cover ~80% of CGL puzzles.
Shortcut tricks
⚡ Two-row confirmation
Never answer from one fitted row. Write the candidate rule symbolically ('c = a×b + 1'), check it on the second complete row, then apply. Ten seconds of checking prevents the most common wrong answer in this topic — a rule that fits only the first row.
Example: Q. Row 1: 3 4 13; Row 2: 5 2 11; Row 3: 6 3 ?
Sol. Guess c = a×b + 1: check row 2: 5×2 + 1 = 11 ✓. Apply: 6×3 + 1 = 19.
Guess → confirm on the spare row → apply. In that order, every time.
Where students lose marks
Fitting a rule to the first row only and answering immediately.
Assuming left-to-right reading of the figure when the rule runs along columns or diagonals.
Over-complicated rules (three operations) when a simple one fits both rows.
Practice sets — 12 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 12 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.