Matrix reasoning tasks: the five rules and how to work them

A matrix task always looks the same: a grid of three rows and three columns, eight cells carrying a pattern, the ninth at the bottom right empty. Below it stand four to six suggestions, and exactly one of them belongs in the gap. Anyone working through an aptitude test, a recruitment test or an intelligence test meets this task type sooner or later. It is the most widespread way of capturing reasoning without measuring language or school content along with it.

This text explains what sits inside such a task: which rule types the writers use, in which order to work through a matrix, and how much practice actually buys you. You can try it afterwards directly in the test, which contains six figural matrices among 30 tasks.

What is actually being tested in that empty cell

The task looks like a spot-the-difference picture but is not one. Rules hold between the eight visible cells, and those rules are the real task. One feature grows from left to right. A second stays constant within the row. A third is spread across the grid so that every value appears exactly once in every row. Anyone who has found all the rules in force can describe the missing cell before even reading the answer options.

That is precisely the object of the test: not knowledge, not vocabulary, not arithmetic skill, but the ability to find an order in unfamiliar material and apply it to a new case. It is why the task still works when somebody speaks a different first language or has not been in a classroom for twenty years.

Why this particular task type turns up everywhere

Test writers face a contradiction. They want to measure reasoning, but every task that works with language or numbers inevitably measures how much somebody has learnt as well. A word analogy presupposes vocabulary. A number series presupposes arithmetic routine. A word problem presupposes reading speed.

The matrix gets around that. It needs no language, no technical vocabulary and no curriculum. It consists of shapes that nobody has to have learnt beforehand. John Raven developed this task format in the 1930s for exactly that reason, and it is still in use across language borders and across decades. What a result from it means, and how far it is comparable across cultures and periods, Raven himself worked through in detail in 2000.

The price of that purity is that the task looks odd at first sight. Somebody seeing it for the first time loses time to the format instead of to the thinking. We come back to that further down.

The rule types a matrix can be solved by

Carpenter, Just and Shell investigated in 1990 what actually happens when people solve these tasks, and identified a manageable number of rule types from which practically every matrix is assembled: a feature stays constant within the row, a feature increases step by step, values are distributed across the grid, and two cells combine into a third. Our own tasks are produced by a program built on the same logic, which additionally treats rotation as a type of its own.

Progression

The most frequent and the easiest type. A feature changes step by step in one direction: the number of dots rises from one to two to three, a line gets longer, a circle gets bigger, an area fills up further. Progressions are easy because the eye notices change by itself.

It only gets difficult when two progressions run at the same time and point in different directions: the dots multiply while the figure shrinks. Then the only thing that helps is counting the two features separately instead of comparing the whole picture. In our tasks, difficulty rises essentially with the number of rules acting at once, and progression is the type that most often mixes with others.

Distribution of three values

This type works like a Latin square: there are three variants of a feature, say three shapes or three fills, and each appears exactly once in every row and exactly once in every column. There is no order and no direction, only completeness.

The check is correspondingly simple. You look at the last row, establish which two of the three values are already there, and you know which one is missing. The mistake many people make is searching for an order that does not exist. If a feature moves in no recognisable direction but nevertheless looks different three times in every row, you are dealing with this type.

Constancy within a row or column

The most inconspicuous type, and therefore the one most often overlooked. A feature simply does not change within the row: all three cells share the same border colour, the same basic shape, the same line weight. Between rows, however, that same feature may well change.

Constancy is rarely the only rule in a task, but it is often the rule that separates two otherwise equivalent answer options. If you are still hesitating between two options after working through all the progressions, it is almost always worth asking: is there a feature that ought to stay the same within the row, and does it do so in both candidates?

Rotation

The same figure appears in every cell, but each time turned by a fixed angle, typically 45 or 90 degrees. The rotation can run along the rows, down the columns, or in both directions at once.

These tasks put more load on spatial imagination than the other types, and they are the reason some people do noticeably differently on figural matrices than on numbers or words. A practical trick: find an asymmetric detail on the figure, a single spike or a dot at the edge, and follow only that detail along the row. It is considerably easier to track the travel of one dot than to rotate a whole figure in your head.

Logical combination between the columns

The most demanding type. Here the third cell of a row arises from the first two according to a fixed combination rule. Either the elements are added, in which case the third cell contains everything from the first two. Or they are subtracted, in which case only what is in cell one and not in cell two remains. Or the rule is that whatever appears twice disappears.

You recognise the type by the fact that the third cell becomes neither bigger nor smaller than its neighbours, but differently composed. The check consists of laying the first two cells mentally on top of each other and seeing what has survived into the third. Once you have read the combination rule off the first row, you apply it to the third.

The method: row first, then column, then the options

There is an order that works reliably on these tasks, and it has three steps.

First: go through the top row from left to right and check one feature at a time in isolation. Number. Size. Shape. Fill. Orientation. Position. Note in your head every rule you find, and deliberately do not look at the picture as a whole.

Second: check the same rules against the columns. A rule that holds in the row and breaks in the column has been misread. A rule that holds in both directions is secure.

Third: describe the missing cell in words before you look down. Something like: three dots, square, unfilled, rotated by 90 degrees. Only then do you search the answer options for the one that matches your description.

Why looking at the options first hurts

The answer options are not neutral. Whoever builds such a task constructs the wrong options deliberately as plausible near-solutions: one agrees on every feature except a single one, one repeats the last visible cell, one carries the progression a step too far.

If you look down first, your mind stops working on the rule from that moment on and starts comparing the suggestions with each other. And comparing similar pictures leads reliably to the option that looks most familiar, not to the one that is right. That is why writing your own description before the first glance downwards produces more correct answers than any other single measure. It costs five seconds and switches the distractors off.

What practice buys and what it does not

This is where too much is usually promised. The effect of practice on matrix tasks splits into two parts, and the two parts behave completely differently.

The first part is familiarity with the task format, and it works immediately and clearly. Somebody who sees a 3x3 grid with a gap for the first time spends the first half minute working out what is being asked at all. That time is missing at the end, and nervousness comes on top. A single practice run largely removes that disadvantage.

The second part is the underlying reasoning ability, and that barely changes through practice. The pooled analysis by te Nijenhuis and colleagues from 2007 summarises many retest studies and concludes that the point gains on repetition do not reflect the same general ability that the first run captures. Hayes, Petrov and Sederberg looked more closely in 2015 and showed that what mainly changes with practice is the solving strategy: practised people scan the grid more systematically.

What follows from that for you

The conclusion is uncomfortable but usable: practice is worth it, but only up to a point, and that point is reached early.

One or two complete runs under realistic circumstances collect the format advantage, which is what this is about. After that the gain flattens off quickly, and what still accrues is routine in handling a particular task format rather than a change in ability. Anybody who buys a preparation course that drills matrices beyond that point is essentially buying routine. That may still be worth something for the nerves before an appointment. It is just a different thing from what is written on the packet.

For us that is the reason the test is free: the format advantage should not cost anything.

Where you meet matrices in Switzerland

For a reader who did not grow up here, this section needs a paragraph of background first, because it explains why these tests carry a weight in Switzerland that they do not carry in the UK or the USA.

After lower secondary school, young people in Switzerland take one of two main routes. One leads to the Gymnasium, the academic upper secondary school that ends with the Matura and gives direct access to university. The other, and the one about two thirds of school leavers take, is the Lehre: a vocational apprenticeship of three or four years leading to a federal certificate of competence, the EFZ, which combines paid work in a training company with school. The apprenticeship is not a fallback for weaker pupils. It leads to protected professions, it can be combined with the Berufsmaturität, the vocational baccalaureate, and from there to the universities of applied sciences and, via the supplementary Passerelle exam, to the regular universities. The figures show the balance: in 2023, 23 per cent of a year group reached a gymnasial Matura and 15.5 per cent a Berufsmaturität.

The consequence for a 14-year-old is that applications for an apprenticeship start in class 8, at roughly age 14, and that many training companies ask for a standardised aptitude test with the application. That is the point at which matrix reasoning enters an ordinary Swiss school career. Note also that education is a cantonal matter: Switzerland has 26 cantons, each with its own school system, and which tests a school uses and when depends on where you live.

Multicheck and Basic Check are commercial aptitude tests taken when applying for an apprenticeship. Both come from the same provider, gateway solutions ag, which operates them under the gateway.one brand, and both are sat at a test centre rather than at home, under supervision and with a fixed appointment. The Basic Check is the successor to the Multicheck Gewerbe and starts in May 2026; its base module covers German, mathematics and cognitive abilities, with further modules such as English, French, practical problem solving, technical understanding and digital skills available on top. That cognitive component is where a language-free reasoning section would sit. Which tasks exactly it contains, the provider does not publish, and we therefore do not claim it.

Stellwerk is a different animal and worth keeping apart from the two above. It is administered by the school, not bought by the applicant, it is run by the teaching materials publisher of the canton of St. Gallen, and it tests school content against Lehrplan 21, the common curriculum of the German-speaking cantons. It is adaptive: it adjusts the difficulty to the answers given and reports on a scale from 200 to 800. There is no figural reasoning section in it at all, which is also why CHF 10.00 per pupil buys something entirely different from CHF 100 at a test centre.

Mensa Switzerland, the Swiss branch of the international high-IQ society, is the third place a Swiss reader meets this task type, and the only one where it is openly the whole point. Its non-verbal admission test works purely with figural material and costs CHF 80.00. It is supervised, which is what distinguishes it from anything taken at home, our own test included.

The defensible statement is this: wherever a procedure contains a language-free section on logical reasoning, that section is with high probability a matrix task or a close relative of one, because there is hardly another task format that works without language and without school content. How the Swiss procedures otherwise differ is set out in our overview of the aptitude tests for an apprenticeship; why the Stellwerk test follows entirely different rules is explained under Stellwerk 8, and where it is compulsory in our overview by canton.

How to practise sensibly

Do one complete run under real circumstances: the clock running, the phone away, no second attempt at a task. That is the part that genuinely helps.

Afterwards go through the tasks you got wrong and assign each one to one of the five rule types in the table above. Almost everybody has a type that consistently gives them trouble, and it is usually rotation or logical combination. That assignment is the real practice, not the number of tasks worked through.

Our test is free, takes 25 minutes and generates the tasks afresh on every run, so on a repeat you will mostly see different ones. How the value is then calculated, what range it carries and what it does not say is set out openly on the method page. The result is an orientation value from a short online test and not a basis for a decision about school or work.