Matrix and grid completion (2x2, 3x3) with rule inference across rows and columns
What this domain tests
A grid of figures with one cell empty. You must find the rule that generates the grid and select the figure that completes it. The rule may run along rows, down columns, or both, and each visual attribute may obey its own rule.
Where the rule can live
- Row-wise operation: cell 1 combined with cell 2 gives cell 3, by superposition, cancellation, or arithmetic on counts.
- Column-wise operation: the same, applied downwards.
- Latin-square distribution: each row and each column contains each variant exactly once, so the missing cell is fixed by elimination.
- Global progression: an attribute rises left to right and top to bottom simultaneously.
- Constant attribute per row: all three cells in a row share a property that changes between rows.
Analyse attribute by attribute
Do not look for one rule for the whole figure. Take the number of sides, then shading, then position, then orientation, each separately, and check whether each forms a row rule, a column rule, or a constant. Most 3x3 items are two or three simple rules stacked.
Worked example
A 3x3 grid; each cell holds a polygon with one vertex marked by a dot. Across row 1 the polygons have 3, 4, 5 sides; row 2 has 4, 5, 6; row 3 has 5, 6 and the missing cell. The row rule is therefore plus one side per step, giving 7 sides in the missing cell. Down each column the dot moves one vertex clockwise per step: in column 3 the dot is at vertex 3, then 4, so the missing cell needs vertex 5. The answer is a seven-sided polygon with the dot on the fifth vertex. Any option that fixes only one attribute is a distractor.
Traps that cost marks
- Assuming all rows share one rule when each row has its own but consistent rule.
- Applying a row rule to a cell defined by a column rule, or the reverse.
- Stopping at the first attribute that works and ignoring the rest.
- Overlooking that the outer boundary or frame is itself an element subject to the rule.
- For a 2x2 grid, forgetting that the fourth cell must satisfy both the row and the column reading.