How to solve matrix puzzles

To solve a matrix puzzle, find the rule one property at a time. Look at the top row, list what changes between the cells (shape, count, size, fill, angle, position), name the rule each property follows, check the same rules in the second row, then build the missing cell from the bottom row before you look at the options. Most misses come from reading the options too early.

What is a matrix puzzle?

A matrix puzzle is a grid of figures, usually three by three, with the last cell empty. The figures follow rules across each row and down each column, and you choose the option that completes the grid. Raven’s Progressive Matrices, published in 1938, made the format famous, and most visual IQ tests today use some version of it, ours included.

The format works because it needs no vocabulary and no math. What it asks for is the ability to find a rule in something you have never seen before, which psychologists call fluid reasoning. We cover that difference in fluid vs crystallized intelligence.

What rules do matrix puzzles use?

In 1990, Patricia Carpenter, Marcel Just and Peter Shell studied how people solve the Advanced Progressive Matrices and found that almost every item used a small set of rules. Their list still describes most matrix puzzles:

Rule	What happens across a row	Our pattern type
Constant in a row	A property stays the same in every cell	Appears inside every type
Quantitative progression	A property grows or shrinks by a steady step	Progression
Figure addition or subtraction	The third cell is the first two cells combined or subtracted	Combination
Distribution of three values	Three values each appear once per row	Distribution
Distribution of two values	Two values appear and a third is empty	Not used in our set

Rotation, where a shape turns by a fixed angle, is a progression of angle. We treat it as its own type because the mistakes people make on it are different: they get the angle right and the direction wrong. The rotation guide covers those.

The same study found that the hardest items were not the ones with strange rules. They were the ones with more rules running at once, because each rule has to be held in mind while you check the others.

A five-step method

1. Read only the top row. Ignore the options. List every property of each cell: shape, how many, size, fill, angle, position.
2. Mark what changes. A property that is the same in all three cells is constant. A property that differs needs a rule.
3. Name each rule. Steady change is a progression. Shuffled values that never repeat are a distribution. Marks that merge or disappear are a combination.
4. Check the second row. The rows can start from different values, so check the rule, not the values. If a rule fails here, it was the wrong rule.
5. Build the answer, then find it. Apply every rule to the bottom row and picture the missing cell. Only now look at the options and pick the one that matches it on every property.

Step 5 is the one people skip. The options in a well-made puzzle are built to look right, so starting from them means arguing yourself into a trap.

Why do wrong options look right?

Each wrong option in a matrix puzzle is usually the answer with one or two properties changed. If you were tracking only the shape, an option with the right shape and the wrong fill looks finished.

Our puzzles are built this way on purpose. Every item in our set has two named “distractor axes”, the two properties its wrong options change, and the page of each puzzle tells you which ones they are. In Puzzle #7, Fills and Shapes, the wrong options change the fill or the shape; in Puzzle #13, Shared Marks, they change which marks are kept or add an extra one.

This is why step 1 lists every property. A decoy that changes the property you weren’t checking is invisible until you check it.

Worked example

Take Puzzle #4, Growing Crosses:

• In the top row the cross is the same shape and the same fill in every cell. Those are constant.
• Its size changes: each cell is one size bigger than the one before. That is a progression of one size per step.
• The second row starts from a different size and grows by the same step, so the rule holds.
• The bottom row’s missing cell is one size bigger than its second cell, with the same shape and fill.

One wrong option has the right size but a hollow cross, though every cross in the grid is solid. Two are solid but shrink instead of grow, or jump two sizes at once. The other two make both mistakes. Building the answer first rules out all five before you have to choose.

How do I get faster?

Practice helps with puzzles like these, mostly by teaching you the rules so you spot them sooner. Three habits make the biggest difference:

• Name the rule out loud. “Count goes down by one” is easier to check than a feeling that something is shrinking.
• Review your misses by trap. If you keep missing fills, check fill first for a week.
• Practice the types that combine rules. Most hard puzzles combine two familiar rules rather than using a new one.

What practice doesn’t do is change the ability the puzzles were designed to measure, as far as the research can tell. We go through the studies in does practicing puzzles change your IQ?.

Practice puzzles by type

• Progression puzzles: something changes by the same step.
• Rotation puzzles: a shape turns or moves by a fixed amount.
• Combination puzzles: two cells combine into the third.
• Distribution puzzles: each value appears once per row and column.

All 30 are on the practice puzzles page, with every answer explained.

Sources

• Carpenter, P. A., Just, M. A., & Shell, P. (1990). What one intelligence test measures: a theoretical account of the processing in the Raven Progressive Matrices Test. Psychological Review, 97(3), 404–431. doi:10.1037/0033-295X.97.3.404
• Raven, J. (2000). The Raven’s Progressive Matrices: change and stability over culture and time. Cognitive Psychology, 41(1), 1–48. doi:10.1006/cogp.1999.0735
• The rules, steps and distractor axes of our 30 practice puzzles, published on each puzzle’s page.