Most abstract reasoning practice stops at rotation, movement, counting and colour cycles. Those are rules you can read inside one figure. The eight families below are harder because you cannot: the answer depends on a neighbour, on the figure two steps back, on a count somewhere else in the figure, or on a total that never changes. Learn to spot them and the ordinary series start to feel easy.
EPSO’s abstract reasoning test gives you about a minute per question: in the current AST 3 competition, 10 questions in 10 minutes, with a pass mark shared with numerical reasoning (10 out of the 20 points of the two tests together). For the basics, start with our abstract reasoning guide and the ten rule families; this page goes one level up.
1. Neighbour rules
Each element’s next state depends on the elements beside it, not on itself. A bead turns black when exactly one of its two neighbours is black; a cell turns black when exactly one cell above, below, left or right of it is black. The figure looks as if it is rotating or flickering at random, but nothing moves.
How to spot it: a ring or grid whose pattern changes in a way no rotation explains, and a number of black elements that jumps up and down. The trap: looking for a turn. Work one element at a time from the last figure, always reading the old colours, never the new ones.
2. Two-back recurrences
Each figure is built from the two before it, not just the last one: a count that is the sum of the previous two (3, 2, 5, 7, 12 → 19), or a grid laid over the grid two steps back with matching black cells cancelling to white.
How to spot it: the differences between neighbouring figures look random, but figure 3 can be rebuilt from figures 1 and 2. The trap: trying to read the answer from figure 5 alone. Test the rule on figure 3 first, then apply it to figures 4 and 5.
3. Coupled step sizes
How far one element moves is set by a count somewhere else in the same figure: a needle turns by as many sectors as there are dots, a marker moves by as many places as there are black beads.
How to spot it: an element whose step keeps changing size, next to something you can count. The trap: using the previous figure’s count. Settle the count of the figure you are drawing first, then move the element by that many.
4. Behaviour by kind
Elements of different colours or shapes follow different rules in the same figure. White kites turn while black kites copy a neighbour; a token moves as many stations as it has corners, so a circle never moves.
How to spot it: some elements behave regularly and others do not. Group them by colour or shape and the regularity appears in each group. The trap: forcing one rule onto everything.
5. Rule cycles
The transformation itself takes turns in a fixed order: a mirror, then a colour swap, then a quarter turn, then a mirror again. Each step is easy; the difficulty is knowing which one comes next.
How to spot it: name the move between each pair of figures out loud. If the names repeat in a pattern (mirror, swap, turn, mirror), the next name is the answer. The trap: assuming the last move repeats.
6. Walkers with memory
A walker moves by local rules and leaves traces: it turns right on a white tile and left on a black one, and flips the tile it leaves; or a pen turns whenever the next pin has already been used, even if that stroke has faded from the picture.
How to spot it: one moving element plus cells or marks that change behind it. The trap: re-reading the state from the last figure only. Some rules depend on history, so track the path as you go.
7. Topology
The rule is about how many pieces, crossings, islands, holes or closed loops there are, and positions are noise. Two strokes that meet at a pin are one piece; a notch bitten out of an edge is not a hole; two cells touching only at a corner are separate islands.
How to spot it: the layout changes completely from figure to figure, yet something countable rises or falls by one. The trap: counting the wrong unit, such as chords instead of crossing points.
8. Conserved transfers
Quantities move between containers and the total never changes: the fullest tube pours half into the next one, bowls with three beads share them out, the poorest purse takes a coin from each of the others.
How to spot it: add up the counts in each figure. If the sum is constant, look for the transfer rule. The trap: moving one container before deciding who gives. All transfers in a figure happen at once, based on the previous figure.
Five practice questions
These are the five free questions of our Abstract Reasoning · Focus III card, in the same form as on the card: five figures, a question mark and five options. Give yourself two minutes each before you open the answer. The figures are drawn for this card; the families each question uses are named in the answer.
Show answer
• The beads: a bead is black in the next figure when exactly one of its two neighbours is black now, and white otherwise — its own colour does not count.
• The clasp: it moves clockwise by as many beads as are black in the same figure.
• The pendant: it changes at random and carries no rule.
Working it out: Apply the neighbour rule to figure 5's four black beads and the answer has six black beads, so the clasp moves six beads clockwise from where it sat in figure 5.
Why not the others:
• A turns the whole ring of beads one place instead of applying the neighbour rule, and moves the clasp by the old count.
• C applies the neighbour rule but counts each bead's own colour too, and moves the clasp anticlockwise.
• D moves the clasp by the previous figure's black beads.
• E lets each bead's own colour count as well, so a bead with both neighbours and itself black stays black.
Shortcut: The ring looks as if it is turning, but no bead travels: each one is recomputed from its two neighbours. Do one bead at a time from figure 5, then count the new black beads to move the clasp. The pendant is there to be ignored.
Show answer
• The patches: each figure is the figure before it turned a quarter clockwise, laid over the figure two before; where two black patches meet they cancel to white.
• The turn: the quarter turn is always clockwise.
• The border: grey when an odd number of patches are black, white when the number is even.
Working it out: Turn figure 5 a quarter clockwise and lay it over figure 4: one pair of black patches meets and cancels, which leaves one black patch — an odd number, so the border is grey.
Why not the others:
• A keeps black where two black patches meet instead of cancelling them.
• B overlays the two previous figures without turning the latest one.
• C turns the latest figure anticlockwise before the overlay.
• D overlays without turning and gives the border the wrong parity.
Shortcut: Nothing here can be read from figure 5 alone: every figure needs the two before it. Write the two grids side by side, turn the later one, then cancel. The border is a free check at the end — count the black patches.
Show answer
• The dots: up two, down one, up two, down one — 1, 3, 2, 4, 3 → 5.
• The needle: it turns clockwise by as many sectors as there are dots in the same figure.
• The grey sector: it is the sector the needle pointed at in the figure before.
• The dots' places: their places in the tray change at random and carry no rule.
Working it out: The answer has five dots, so the needle turns five sectors clockwise from where it pointed in figure 5, and the grey sector is the one it pointed at in figure 5.
Why not the others:
• B adds two dots again instead of taking turns, and turns the needle by six.
• C turns the needle by the previous figure's dots and greys the sector it now points at.
• D turns the needle by the previous figure's three dots instead of this figure's five.
• E adds two dots again, turns by six and greys the sector the needle points at now.
Shortcut: Settle the dots first: the needle cannot be checked until you know how many there are. Then count sectors from figure 5's needle, not from the grey sector.
Show answer
• The distance: every token moves as many stations as it has corners — so the circle never moves.
• The direction: odd numbers of corners go clockwise, even numbers anticlockwise.
• The shading: a token is black while it stands on the top straight and white anywhere else.
Working it out: From figure 5 the triangle moves three stations clockwise, the square four anticlockwise and the pentagon five clockwise; the circle stays put. The triangle, the square and the pentagon all land on the top straight, so all three are black.
Why not the others:
• A moves the pentagon four stations instead of five.
• B sends the square clockwise and shades by the old places.
• C moves every token correctly but shades each one by where it stood before moving.
• D sends the square clockwise.
Shortcut: The circle is the clue: it never moves because it has no corners. Count corners, not frames — and shade only after every token has landed.
Show answer
• The turn: on a white tile the walker turns right, on a black tile it turns left, then it steps one tile forward.
• The tiles: the tile it steps off changes colour; no other tile ever changes.
• The pebble: the grey pebble wanders at random and carries no rule.
Working it out: In figure 5 the walker stands on a white tile, so it turns right and steps one tile forward; the tile it leaves turns black.
Why not the others:
• A turns the wrong way and changes the tile it arrives on instead of the one it left.
• B turns the wrong way and leaves every tile as it was.
• C turns the wrong way for the colour of the tile it stood on.
• E steps two tiles and leaves every tile as it was.
Shortcut: Read the tile UNDER the walker in figure 5 — its colour gives the turn — and change that tile, not the next one. The pebble is there to be ignored.
These questions are written by EU Testing in the EPSO format. They are not official EPSO questions.
How to use these families in the real test
In the test you will rarely know which family you are facing. A quick order of checks helps: first look for anything constant (a total, a number of pieces); then count things in each figure; then compare each figure with the one two steps back; then group the elements by kind. If none of that works, name the move between each pair of figures and look for a cycle. Each check takes a few seconds, and one of them usually breaks the series open.
55 more questions on these families
Abstract Reasoning · Focus III: 60 figure sequences on these eight families, every one with a worked answer, in 13 languages.
Open Focus III → First 5 questions free · the rest with PremiumFrequently asked questions
Does EPSO really use rules this hard?
EPSO does not publish its questions beyond a few samples, which use plain rules such as rotation and colour changes. These families go further on purpose: practising on harder series makes the test’s own rules quicker to see, and the habits they train, such as counting, checking totals and comparing figures two steps apart, apply to any series.
How much time do I have per abstract reasoning question?
About a minute. In the current AST 3 competition, EPSO/AST/158/26, the abstract reasoning test has 10 questions in 10 minutes, and its pass mark is shared with numerical reasoning: 10 out of the 20 points of the two tests together.
What is the fastest way to find the rule?
Check for something constant first, such as a total or a number of pieces, then count elements in each figure, then compare each figure with the one two steps back, then group elements by colour or shape. Naming the move between each pair of figures reveals rule cycles.
Where can I practise more of these?
Our Abstract Reasoning · Focus III card has 60 sequences built on these eight families, with a worked answer for each. The first five, the ones on this page, are free.
Practice questions from the EU Testing Abstract Reasoning · Focus III card; test format from the notice of EPSO/AST/158/26. Published 12 October 2026.