How to solve matrix puzzles

A matrix puzzle shows a 3×3 grid of cells with the last cell missing. Each cell holds one or more figures, and the figures change from cell to cell according to a few rules. Your job is to work out the rules and pick the figure that completes the last row.

Matrix puzzles have been used in reasoning tests since the 1930s because they need no language and depend mostly on spotting rules, not on what you have learned. The puzzles on this site are generated by our own code; the examples below describe the kinds of rule they use.

The features that can change

Every figure is described by a handful of features:

• Shape: circle, square, triangle, diamond, pentagon or cross.
• Number: one to four copies of the shape.
• Size: small, medium or large.
• Fill: empty, solid, striped or dotted. Colour never matters.
• Turn: in steps of 45°. A small dot on each figure shows which way is “up”.
• Edge marks: little bars and corners on the cell’s edges.

The five rules

1. Constant

A feature stays the same along each row. If every cell in the first row has striped figures, expect the last row to keep its own fill too.

2. Progression

A feature changes by the same step from cell to cell: one more shape each time, a step bigger, or a turn of 45° clockwise. Read the step from the first two rows, then apply it to the last.

3. Distribution of three

Each row contains the same three values of a feature in a different order: for example a circle, a square and a triangle. The missing cell gets the value that row has not used yet.

4. Addition

The third cell combines the first two. For edge marks, every mark that appears in the first or the second cell also appears in the third.

5. Exclusive or (XOR)

The third cell keeps only the marks that appear in exactly one of the first two. Marks that appear in both cells cancel out. This is the rule people miss most often, because at first it looks like addition.

Harder puzzles combine two or three rules on different features, and the hardest ones add a feature that changes at random and does not matter at all.

A strategy that works

1. Go feature by feature. Look only at the number of shapes across the first row, then the second. Is it constant, rising or a set of three? Then move on to size, fill, shape, turn and marks.
2. Use two rows, not one. A rule should explain both of the first two rows. If it only fits one, it is the wrong rule.
3. Predict before you look. Decide what the missing cell should contain before looking at the options. That protects you from attractive wrong answers.
4. Eliminate. Cross out every option that breaks any one rule. The answer is the one that survives every check.

The traps behind wrong answers

Wrong options are rarely random. Each one is built from a specific mistake:

• A rule ignored: one feature is copied from the cell to the left.
• The wrong step: a progression applied one step too far.
• Addition instead of XOR, or the reverse, for edge marks.
• One feature swapped: the right figure with a single detail changed.

The options are also balanced so that no feature value is more common in the correct answer than in the wrong ones. Guessing “the most typical-looking option” does not help; only finding the rules does.

Practise

The best way to get comfortable is a few untimed puzzles with feedback. Practise matrix puzzles for free: after each answer you see which option was right and why. When you are ready, take the quick test.