Missing Number

The CGL workhorse: grids where rows (or columns) share a rule

The format: a 3×3 grid — sometimes two complete grids and a third with a hole. Each row obeys the same rule turning its first two numbers into its third. Everything is row-arithmetic; the skill is finding the rule fast.

The two-number bridge. Cover the third column and hunt a bridge from column 1 to column 2 — or from columns 1+2 to column 3. The classic bridges, in testing order:

• $c = a + b$ and $c = a - b$ (or $|a-b|$) — check signs.
• $c = a \times b$; then $c = 2ab$, $c = ab + k$, $c = ab - k$ with a small constant.
• $c = a^2 + b$, $c = b^2 + a$, $c = a^2 + b^2$ — the Pythagorean look (3-4-25, 5-12-169).
• $c = (a + b)^2$ or $(a - b)^2$ — results like 49, 64, 144 are squares.
• Digit tricks (see the digit-operations lesson): digit sums and reversals.

Confirm on every row. Compute the candidate rule on all complete rows. A rule fitting row 1 and row 3 but not row 2 is dead — keep hunting. This one habit separates solvers from guessers.

Column rules. When rows refuse every bridge, the rule runs down the columns: third row = f(first row, second row) cell by cell, or column 2 = average of columns 1 and 3, or column 3 = column 1 − column 2. Same procedure, transposed.

Two complete grids. Solve the left grid's rule first (it has no hole), verify it reproduces the middle grid, then apply to the right one. The rule never changes between grids; only the numbers do.

Circle and triangle figures. Match by position: opposite segments of a circle (constant sum or product), corners vs centre of a triangle (centre = sum or product of corners), inner ring vs outer ring. Write the matched pairs as little equations and look for the common thread.

Worked walk-through. Rows (4, 6, 10), (5, 3, 22), (7, 2, ?). Sum: 4 + 6 = 10 ✓ on row 1 — tempting — but 5 + 3 = 8 ✗ on row 2, so the sum rule dies. Product: 24 ✗. Square family: $a^2 - b$ gives 16 − 6 = 10 ✓ and 25 − 3 = 22 ✓. Only now apply it: 49 − 2 = 47. Notice the trap: had you stopped at row 1, you would have answered 7 + 2 = 9. The discipline is the same every time: one candidate, all rows, next candidate.

Speed tricks. Keep squares to 25 and cubes to 12 memorised; spot $a^2 - b^2 = (a+b)(a-b)$ to avoid big squaring; and when third cells are all odd while inputs are mixed parity, product ± 1 is a strong suspect.

Quick revision

• Bridge hunt order: + / − → × / ÷ → ×2, ab ± k → a²+b, b²+a, a²+b² → (a±b)².
• The rule must reproduce EVERY complete row (and grid 2 in two-grid puzzles).
• Rows stuck → test columns; grids stuck → test opposite segments / centre vs corners.
• Squares in the options (25, 49, 144) beg for a square-based rule.
• Digit sums and reversals are fair game — see digit-operations.