Matrix puzzles — a 3 × 3 grid of shapes with one cell missing — are the classic non-verbal IQ question. They look endless, but most are built from a short list of rules. The researchers behind the open item bank our IQ Test uses designed every puzzle from just six construction rules, often several at once. Learn to recognise them and the puzzles stop looking random.
How to read a matrix
Rules almost always run along the rows: the third cell of each row follows from the first two. Use the first two rows to work out the rule, check that it holds for both, then apply it to the bottom row to build the missing cell. If a rule doesn't fit a row, it isn't the rule.
It also helps to treat each type of piece separately. In our test the pieces fall into five families — corner triangles, diagonal lines, small edge bars, a centre square or circle, and small triangles at the edges — and each family can follow its own rule in the same puzzle.
Rule 1: Addition
Everything in the first cell plus everything in the second cell appears in the third.
Example: cell 1 has the top-left corner, cell 2 the bottom-right corner, cell 3 has both.
Rule 2: Subtraction
The pieces in the second cell are removed from the first.
Example: cell 1 has all four corners, cell 2 has the two top corners, cell 3 has only the two bottom corners.
Rule 3: Disjunctive union ("keep the differences")
Pieces that appear in both the first and second cell disappear; pieces in only one of them stay. Programmers know this as XOR.
Example: cell 1 has the left and top bars, cell 2 the top and right bars. The top bar is in both, so cell 3 has the left and right bars.
This is the rule people miss most often: in the bank's data, single-rule puzzles of this type were solved by about 60% of test-takers, against over 80% for addition, subtraction and intersection.
Rule 4: Intersection ("keep what's shared")
The opposite of rule 3: only pieces that appear in both cells survive.
Example: cell 1 has a circle and two lines, cell 2 has the circle and one other line. Cell 3 has just the circle.
Rule 5: Rotation
A piece moves around the cell, clockwise or anticlockwise, one step per column.
Example: a small triangle sits at the top, then on the right, then at the bottom.
Rule 6: Completeness
Each row contains the same set of pieces, just arranged differently — like a Sudoku row. Find which member of the set is missing from the bottom row.
Example: every row has a solid square, an empty square and a solid circle once each; the bottom row already shows two of them.
Why harder puzzles feel so much harder
Difficulty rises mainly with the number of rules combined. In the calibration data, puzzles with one rule were solved by about 81% of people on average, those with three rules by 63%, and those with five rules by only 38%. Each extra rule is another family of pieces to track at the same time.
A step-by-step method
- Scan the first row. What appears, disappears or moves between cells?
- Split by family. Ask the question separately for corners, lines, bars, centre shapes and small triangles.
- Name a rule for each family — addition, subtraction, XOR, intersection, rotation or completeness — or "stays the same".
- Check the second row. Every rule must hold there too; if not, try another.
- Build the answer for the bottom row family by family, then compare the whole cell with the row once more before confirming.
Does practising raise your IQ?
Practice makes you faster and more familiar with the format, so scores on this kind of test usually go up a little. That reflects familiarity more than a change in underlying ability — which is why the first attempt at a test is the fairest measure.
FAQ
Do rules ever run down the columns? In some tests, yes. If no row rule fits, try the columns.
Can one puzzle use several rules? Yes — up to five in the open item bank, one per family of pieces or combined.
Where can I practise? The IQ Test page starts with two practice puzzles that show the correct answer, and the review at the end shows your cell next to the correct one for every puzzle.
Try it now
Put the six rules to work in the free adaptive IQ test. It adapts to your level, so you will always get puzzles that stretch you.