Matrix Logic

What you see

The screen shows a 3 by 3 matrix of panels. The bottom-right panel is missing and marked with a question mark. Below it sits a row of options: four at the lowest levels, five in the middle, six at the top.

Each panel is itself a small 3 by 3 grid of cells. Some cells hold a symbol, some are empty. Within a panel every symbol is identical, so a panel has exactly three properties:

• Cells: which of its nine positions are occupied (always between three and six in the first two columns).
• Shape: circle, square, triangle or diamond.
• Fill: outline, striped or solid.

You choose an option by clicking it or pressing its number key; the chosen panel appears in the empty slot, and choosing another replaces it. When you are happy, press Check answer (or Enter). The verdict appears straight away with the cell rule every row follows, and the board marks the right option in green, and your choice in red if it was a different one. A clock counts up while you think. There is no cut-off in practice sessions, but your time is part of the score, as explained below.

How a puzzle is built

Every trial draws one rule for each property, and every row obeys all three.

1. A cell rule decides which cells are occupied. There are seven. Four are binary: panel 3 is built from panels 1 and 2 by a set operation. Three are unary: panel 2 is panel 1 transformed, and panel 3 is panel 2 transformed the same way again.
2. A shape rule: either the shape stays the same across the row, or it cycles one step per column through circle, square, triangle, diamond and back to circle.
3. A fill rule: the fill stays the same, cycles one step per column through outline, striped, solid and back, or inverts (outline and solid swap in the middle column, and striped stays striped).

Each row starts from its own random panels, shape and fill. Only the rules are shared. That is why the three rows can look completely different and still be one puzzle.

The generator also guarantees something useful: it checks every combination of rules in its vocabulary against the eight visible panels, and throws the puzzle away if two different combinations that fit would predict different answers. There is always exactly one defensible answer, and exactly one option matches it.

What changes as levels rise

Three things climb together. The number of simultaneous rules (properties that change across a row) goes from one at level 1 to three at levels 9 and 10. The cell rules shift from the easy, visible ones (union, mirror) to the ones that remove cells (xor, subtract). And from level 5 the wrong options get nastier: when the cell rule is a set operation, the first distractors are what the other set operations would have produced from row 3. At levels 1 to 4 the distractors are simpler near-misses: the answer rotated or mirrored, the right cells with the next shape or next fill, or the answer with one cell toggled.

What it measures

Matrix Logic is an inductive reasoning task built on abstract transformation. You infer rules from two complete rows, then apply them to the third. It feeds the pattern recognition facet of your reasoning score and carries the largest single weight in that dimension, 15 per cent.

Its ancestor is Raven's Progressive Matrices, first published by John C. Raven in 1938. The Standard Progressive Matrices has 60 items in five sets of twelve; the Advanced version has 48. It was built to measure the eductive, or meaning-making, component of Spearman's general factor, and it remains widely used in both applied and research settings as a measure of fluid reasoning.

The most influential analysis of how people solve these items is Carpenter, Just and Shell. They found that a handful of rule types describes almost all Raven items: a value constant across a row, a quantitative progression, figure addition or subtraction, and distributions of values across a row. Later work split figure addition into addition, "common" (keep what both share) and "unique" (keep what only one has). Matrix Logic's vocabulary maps closely onto theirs:

The same paper identified what separates strong and weak solvers: less the ability to see a rule than the ability to hold several rules and their sub-goals in order. That is exactly the load Matrix Logic raises as the levels climb.

How it is scored here

Each trial is right or wrong: the chosen option either is the generated answer or is not. There is no partial credit for a panel that gets two of three properties right.

That accuracy is combined with a small speed term. The app estimates how long a trial of this difficulty should take (for reasoning games, 6 seconds at the easiest difficulty, plus 2.4 seconds per step). Accuracy carries 85 per cent of the weight and speed relative to that estimate carries 15 per cent, with speed capped at three times faster or slower. Taking three times longer than expected on a correct answer costs about 15 per cent of that trial's value; a wrong answer costs all of it.

Each trial's value is then multiplied by its measured difficulty, which for Matrix Logic comes mainly from the number of simultaneous rules (2.5 points each), plus half a point for each option beyond four, plus up to two points when several rule combinations fit the matrix. Your game score compares this attainment with a reference: 50 means you are performing at the level the difficulty you work at predicts. Recent sessions count more than old ones, and no score appears until you have enough trials for it to mean something.

Difficulty adapts during a session. You start at level 3; each correct answer moves the next trial up one level and each wrong answer or skip moves it down one. A skip scores zero, so skipping a hard trial never helps.

Your profile also shows diagnostic numbers for this game: the most rules you have handled at once, how long you take to see the rule, how much extra time each added rule costs you, and what share of your errors went to close look-alikes rather than random options. The first trial of each session is excluded from timing measures.

Where you will meet it

Figure matrices and series appear in almost every non-verbal reasoning section.

AMCAT's inductive topics include analogy and classification pattern recognition, which use the same comparisons, and prep sites report figure-series items in the abstract reasoning sections of several campus tests. Our game is not a copy of any of these; it is the same family of task, with the rule vocabulary made explicit so you can practise the method.

The method

1. Cover the options

   Look only at the matrix first. This is constructive matching, and it is the strategy with the strongest evidence: people instructed to study the matrix and imagine the answer before seeing the options scored 12.77 of 18 on half of Raven's Advanced Matrices, against 10.45 for people left to their own approach.

2. Decide unary or binary from row 1

   Compare panels 1 and 2. If panel 2 has the same number of symbols and looks like panel 1 turned, flipped or slid, the cell rule is unary. If the counts differ, or panel 2 has no geometric relation to panel 1, it is binary. Unary rules never change the count.

3. Pin the cell rule

   For a unary rule, check which transformation turns panel 1 into panel 2, then confirm that the same transformation turns panel 2 into panel 3. For a binary rule, use the three-question test in the table below. Confirm on row 2 before you trust it.

4. Pin the shape rule

   Same shape across each row, or one step along circle, square, triangle, diamond? Check both complete rows.

5. Pin the fill rule

   Same fill, one step along outline, striped, solid, or outline and solid swapping? A striped row tells you nothing about invert, so use the other rows.

6. Build the missing panel, then find it

   Apply all three rules to row 3. Say it in words: "diamonds, outline, top-centre, centre and bottom-right". Only now look at the options, and check each property of your match.

The binary test works cell by cell. Pick a cell in panel 3 and ask where it came from:

Two looks usually settle it. A shared cell that survives means union or intersect; one that vanishes means xor or subtract. Then a B-only cell separates the pairs: kept means union or xor, dropped means intersect or subtract.

Worked example 1

Show the solution

Unary or binary? In row 1, panel 1 has three solid squares and so does panel 2, laid out as the same L shape turned. Counts are equal: unary.

Which transformation? The L in the top-left corner has moved to the top-right corner, pointing the other way: a quarter turn clockwise. Panel 3 is the same L turned again, now in the bottom-right corner. Row 2 confirms it with outline triangles. The cell rule is rotate.

Shape and fill. Row 1 is solid squares throughout, row 2 outline triangles throughout. Both are constant.

Build it. Row 3 is striped circles. Panel 1 has the left column plus the bottom-centre cell; panel 2 is that turned a quarter (the top row plus middle-left). One more quarter turn gives the right column plus top-centre: top-centre, top-right, middle-right, bottom-right.

Match. That is option 3. Option 1 turns the wrong amount, option 2 is the answer mirrored, and option 4 has the right cells in the wrong shape.

Worked example 2

Show the solution

Unary or binary? Row 1, panel 1 has four circles and panel 2 has four squares in a different arrangement that is not a turn, flip or slide of panel 1. Binary.

Which set operation? The top-centre and centre cells are in both A and B; neither appears in panel 3, so it is xor or subtract. Middle-right and bottom-right are in B only, and both appear in panel 3. Subtract would have dropped them, so the rule is xor. Row 2 agrees.

Shape. Row 1 runs circle, square, triangle. Row 2 runs triangle, diamond, circle. One step per column: cycle.

Fill. Row 1 is outline, solid, outline; row 2 is solid, outline, solid. Not a cycle (that would pass through striped). It is invert.

Build it. Row 3 has outline squares at top-left, centre, middle-right and bottom-right, then solid triangles at top-left, top-centre and middle-right. XOR removes the shared top-left and middle-right, keeps centre and bottom-right from A, and top-centre from B. The shape steps on from triangle to diamond; the fill swaps back to outline. The answer is outline diamonds at top-centre, centre and bottom-right.

Match. That is option 3. Look at what the others are: option 1 is the union, option 4 the intersection, option 5 the subtraction, and option 2 has the right cells as circles. At the tight levels, the wrong options are the other rules' answers. If you have not pinned the set operation, one of them will look right.

Tips and traps

Levelling up

The first big jump is level 5, where the shape rule may start cycling and the distractors become other rules' answers. The second jump is level 7, where the shape always cycles and fills may start cycling too, so you are tracking three rules on most trials. At levels 9 and 10 every trial has all three rules changing and six options.

Adapt in three ways as you climb:

1. Write nothing down, but say the rules. A three-word summary ("xor, cycle, invert") keeps the rules in working memory while you build the answer.
2. Separate the properties. Build the cells first, then attach shape and fill. Options that are right on cells but wrong on shape are common, and easy to spot once you look for them.
3. Spend time where it pays. At high levels a correct slow answer is worth far more than a fast wrong one. Speed only carries 15 per cent of the weight.

1. Week 1
   • Three sessions of 10 to 15 trials, options covered until you have named the cell rule
   • Learn the binary table until you can run it without looking
   • After each miss, name which rule you got wrong
2. Week 2
   • Aim to hold level 5 and above: name all three rules before choosing
   • Practise counting symbols first on every trial
   • Note whether your misses are on cells, shape or fill
3. Week 3
   • Push towards levels 7 to 8, where three rules change together
   • Check each option property by property before confirming
   • Mix in one RuleMatch session for figural variety
4. Week 4
   • Timed sets: keep accuracy above 80 per cent while bringing your average time down
   • Review the session report for errors on close look-alikes
   • Rest the day before any real test

Check yourself

6 practice questions

1. In row 1, panel 1 has five symbols and panel 2 has three, in an unrelated layout. What can you conclude?
2. A cell appears in both panel 1 and panel 2 of a row, and is missing from panel 3. Which cell rules are still possible?
3. Under the mirror rule, how does panel 3 relate to panel 1?

and 3 more

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Sources

1. Raven's Progressive Matrices. Wikipedia. en.wikipedia.org/wiki/Raven's_Progressive_Matrices
2. 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
3. Verguts, T., & De Boeck, P. (2002). The induction of solution rules in Raven's Progressive Matrices Test. European Journal of Cognitive Psychology, 14(4), 521–547. doi:10.1080/09541440143000230
4. Gonthier, C., & Thomassin, N. (2015). Strategy use fully mediates the relationship between working memory capacity and performance on Raven's matrices. Journal of Experimental Psychology: General, 144(5), 916–924. doi:10.1037/xge0000101
5. SHL. Cognitive assessments: Verify Interactive Inductive Reasoning. shl.com
6. Aon. Prepare for your online assessment: logic tests. aon.com
7. TCS iON. National Qualifier Test: cognitive assessment sections. tcsion.com
8. AMCAT. AMCAT syllabus: Logical Ability. myamcat.com/amcat-syllabus

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