Guide
Matrix reasoning, explained
A matrix puzzle shows a 3x3 grid of patterns with the last cell missing. Your job is to work out the rules that run across the rows and down the columns, then pick the piece that fits. It looks simple. It is one of the most informative puzzle types in all of intelligence testing.
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What a matrix puzzle looks like
Picture a 3x3 grid. Each cell holds a small arrangement of shapes: perhaps one circle, then two, then three across the top row. The second and third rows follow the same logic with different shapes or a different starting point. The bottom-right cell is empty. Below the grid sit several options, and exactly one of them completes the pattern in every direction.
The format comes from Raven's Progressive Matrices, developed by John C. Raven in the 1930s and first published in 1938. They are nonverbal and are widely used as a measure of fluid, culture-reduced reasoning (Wikipedia: Raven's Progressive Matrices): reasoning with new material rather than drawing on what you already know (Wikipedia: Fluid and crystallized intelligence). There is nothing to memorize and nothing to read, which is why matrix puzzles sit at the center of nonverbal testing.
The rules matrix puzzles use
Almost every matrix puzzle is built from a short list of rule types. Learning to name them is the single most useful thing you can do before a test, because it turns a confusing picture into a checklist.
- Count. The number of elements goes up or down by a fixed step: one, two, three.
- Shape. The kind of shape changes in a fixed order, for example circle, square, triangle.
- Size. Elements grow or shrink step by step.
- Rotation. A shape turns by a fixed angle in each cell, such as a quarter turn.
- Fill. Shapes cycle through solid, outlined and striped.
- Position. An element moves across the cell, for example from left to center to right.
- Distribution of three. Each row contains the same three values, such as one circle, one square and one triangle, in a different order. Nothing “progresses”; each value simply appears once per row.
- Overlay. The third cell in a row is made by combining the first two. Sometimes it keeps every line from both, sometimes only the lines they share, and sometimes only the lines that appear in exactly one of them.
Easy puzzles use one rule across the rows. Harder ones combine two rules, one running across the rows and another running down the columns. The hardest combine three rules or use an overlay, where you have to compare cells line by line.
Try one · Abstract Reasoning
Which tile completes the grid?
Tap the option you think is right, or show the answer.
Show the answer
The missing tile has a large hatched square (option 7).
How to solve it
- Start with the first column and name what changes from tile to tile.
- Every row and every column contains each shape (square, cross and diamond) exactly once.
- Every row and every column contains each fill (outline, hatched and solid) exactly once.
- The size (large) stays the same in every tile.
- Check the rules on the second column, then apply them to the third.
- So the missing tile has a large hatched square.
That was 1 puzzle. The full test has 25.
A method that works
- Read the top row on its own. Ask what changes from the first cell to the second, and from the second to the third. Say it in words: “one more dot each time”, “turns a quarter clockwise”.
- Check the second row. The same rule should hold, possibly starting from a different value. If it does not, your description is wrong or incomplete. Go back.
- Check the columns. A second rule often runs top to bottom. Count, shape and fill are common column rules.
- Try the special cases. If nothing progresses, test distribution of three. If the third cell looks like a mix of the first two, test an overlay.
- Predict before you look. Describe the missing cell completely: what shape, how many, what size, which way it faces, what fill. Then scan the options for that exact description.
- Verify against every rule. When an option looks right, check it against each rule you found, not just the most obvious one.
The traps
Wrong options are not random. In a well-built matrix test, most of them are made by taking the correct answer and breaking exactly one rule: the right shape with the wrong count, the right count with the wrong rotation. That is deliberate. If you only check the most noticeable rule, two or three options will look correct, and you will pick one of them more or less by chance.
Two other traps are common:
- The repeat. An option that copies a cell already in the grid. It feels familiar, which is exactly why it is there.
- The near twin. An option that differs from the answer by one small detail, such as a stripe direction or a dot position. Slow down for the final comparison.
This is also why you cannot reliably find the answer by spotting the “odd one out” among the options. When each wrong option differs from the right one in a different way, the answer is the one that sits in the middle of all of them, and only the rules tell you which that is.
Speed, guessing and practice
Matrix puzzles reward care more than speed. Most mistakes come from stopping after finding one rule. On IQ Octopus, your estimated IQ depends only on which puzzles you answer correctly, so it is worth spending the extra seconds to verify. If you are truly stuck, eliminate the options that clearly break a rule and choose among the rest; an unanswered puzzle counts as wrong.
Practice helps a little. Retaking cognitive tests raises scores by about a quarter of a standard deviation on average (Hausknecht et al., 2007). Most of that is familiarity with the format, which is fair to gain before a test. Our guide on how to prepare covers what is worth doing.
Matrix puzzles on IQ Octopus
The Abstract Reasoning section of IQ Octopus has 9 matrix puzzles, the largest section on the test. They are generated fresh for each attempt from the rule types above, ordered from one rule to three, with eight options each. You can try a worked example on the Abstract Reasoning practice page, and read about the related sequence puzzles in pattern recognition.
Sources
- Raven's Progressive Matrices. Wikipedia. en.wikipedia.org/wiki/Raven%27s_Progressive_Matrices
- Fluid and crystallized intelligence. Wikipedia. en.wikipedia.org/wiki/Fluid_and_crystallized_intelligence
- Hausknecht, J. P., Halpert, J. A., Di Paolo, N. T., & Moriarty Gerrard, M. O. (2007). Journal of Applied Psychology, 92(2), 373-385. doi.org/10.1037/0021-9010.92.2.373