A Matrix Puzzle With One Rule You Can Say Out Loud

A matrix puzzle is a grid with one cell missing, and you solve it by finding the rule that fills the gap from the rows and columns. The one below follows two rules, and its answer is a triangle with 1 dot. A matrix reasoning test is a set of such puzzles that get harder, and one item alone says very little about your IQ.

Matrix reasoning is the item format most people picture when they picture an IQ test: a grid with one cell missing and a row of candidate answers underneath. Test designers like it because it needs little vocabulary or arithmetic, only the ability to work out what rule is generating the pattern.

What is a matrix reasoning test?

A matrix reasoning test is a set of grid puzzles, each with one cell missing, where you pick the option that completes the pattern. The puzzle below is one such item. A full test strings many together, from easy to hard.

The format sits inside several IQ tests. The Wechsler scales include a subtest called Matrix Reasoning, and our guides to the WISC-V index scores and the WAIS-IV and WAIS-5 index scores show where it sits. It is in the Fluid Reasoning index on the WISC-V and WAIS-5, and in Perceptual Reasoning on the WAIS-IV. Pearson's manuals give the item counts, which we have not read, so we print none.

Raven's Progressive Matrices is built entirely from this format. Lionel Penrose and John C. Raven described the matrices in 1936 (British Journal of Medical Psychology). Our guide to Raven's Progressive Matrices scores reports 60 items in five sets of twelve for the Standard version.

People search for a matrix puzzle, matrix puzzles or matrix reasoning puzzles, and others ask what is matrix reasoning. All of them mean this format. Its close relatives are analogy questions, odd-one-out questions and number sequence puzzles.

Can you solve this matrix puzzle?

Here is one you can solve in text. Read the grid, then stop and try it before scrolling on.

Each cell of a three-by-three grid contains one shape, and inside that shape are some dots.

• Top row: a circle with 1 dot, a circle with 2 dots, a circle with 3 dots.
• Middle row: a square with 3 dots, a square with 1 dot, a square with 2 dots.
• Bottom row: a triangle with 2 dots, a triangle with 3 dots, and the missing cell.

The candidate answers are: a triangle with 1 dot, a triangle with 2 dots, a circle with 1 dot, a square with 1 dot, a triangle with 4 dots. Which one belongs in the missing cell?

What is the answer to the matrix puzzle?

The answer is a triangle with 1 dot. Shape is constant along each row, so the missing cell is a triangle, and the dots form a Latin square, so the missing count is 1.

Two features vary independently, so there are two rules and both have to hold.

The shape rule is the easy one. Every cell in the top row is a circle, every cell in the middle row is a square, every cell in the bottom row is a triangle. Shape is constant along each row. The missing cell is in the bottom row, so it is a triangle. That single step eliminates the circle and the square from the options.

The dot rule takes one more moment. Look at the rows: the top row runs 1, 2, 3. The middle row runs 3, 1, 2. The bottom row runs 2, 3, and the unknown. Each row uses the numbers 1, 2 and 3 exactly once — so the missing value must be 1, because 2 and 3 are already used in that row.

Check it against the columns, because a good item is consistent both ways. The first column reads 1, 3, 2. The second reads 2, 1, 3. The third reads 3, 2, and the unknown — and 1 is the only value not yet used in that column. Both rules agree.

The answer is a triangle with 1 dot. The grid is a small Latin square: three values, each appearing exactly once in every row and every column, with a second independent feature layered on top of it.

Why are the other four options wrong?

Each wrong option breaks at least one rule. Here are all five options against the two rules.

• Triangle with 1 dot: shape rule holds, dot rule holds. This is the answer
• Triangle with 2 dots: shape rule holds, dot rule broken, because 2 is already in that row
• Circle with 1 dot: dot rule holds, shape rule broken
• Square with 1 dot: dot rule holds, shape rule broken
• Triangle with 4 dots: shape rule holds, but 4 appears nowhere in the grid

In more detail, each wrong option is built to reward a different mistake.

• The triangle with 2 dots satisfies the shape rule and breaks the dot rule — 2 already appears in that row. It is there to catch anyone who solved half the puzzle.
• The circle with 1 dot satisfies the dot rule and breaks the shape rule. Same trick, other half.
• The square with 1 dot does the same thing with a different shape.
• The triangle with 4 dots introduces a value that appears nowhere in the grid, which is the classic distractor for someone extrapolating a sequence instead of finding a rule.
• Only one option satisfies both rules at once, and that is always how a well-built matrix item is constructed.

Does solving a matrix puzzle say anything about IQ?

Honestly: not much on its own. A single item measures almost nothing. Real matrix tests use many items of graded difficulty, because performance on any one of them is heavily influenced by whether you happened to see the rule quickly, and that varies from item to item even within one person. A longer puzzle that rests on bookkeeping rather than a single rule, the Einstein riddle, is worked through in who owns the fish.

What is worth taking from it is the method. Naming the rule before looking at the options is the single most useful habit on this item type, because the options are written by someone who knows which partial solutions are tempting. Solving half the puzzle and then choosing the answer that fits the half you solved is the most common way to lose an item you were capable of getting. The same row-and-column logic drives Sudoku; see does Sudoku increase IQ for what the studies show.

And the honest caveat about practice: doing a lot of these makes you better at these. That gain is real, it is worth having if you are about to sit an assessment that uses this format, and it is narrower than the phrase "brain training" implies. The evidence that practice on a specific task transfers to general reasoning is much weaker than the evidence that it improves the task you practised. For how much of a full IQ score this item type actually carries, see is IQ just pattern recognition. How large retest gains can be is covered in our guide to the practice effect on cognitive ability tests.

To practise more puzzles of this kind, try our logic puzzles. For a score on the standard scale rather than one puzzle answer, take the IQ Metrics test.

Common questions