Matrix Reasoning IQ Test: 4 Worked Examples

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A four-step method before you choose an answer

1. Describe the features separately. Count the objects, then inspect direction, position, shape and fill. “It looks similar” does not identify a relationship.
2. State one candidate rule. For example, “one circle is added at each step to the right.” A precise statement gives you something to test.
3. Try to disprove it. Check another completed row or column. If the rule predicts a different known cell, reject it instead of changing it just to preserve your first guess.
4. Predict the missing cell before comparing options. Check every relevant feature. An option can have the right direction and the wrong number of objects.

Not every puzzle uses the same relationship across rows and down columns. Check the relationships the given cells support; do not invent a second rule just because the grid has two directions.

Example 1: count the circles and reject a tempting rule

All the circles in this example have the same shape, size and fill.

Choose from A: four circles; B: five circles; C: six circles; D: five triangles. Pause before reading the explanation.

Answer: B, five circles. Each step to the right adds one circle. The same increase also holds down each column, so both checks predict five circles in the missing cell.

The tempting rule “add the first two counts to get the third” fits row 1: 1 + 2 = 3. It fails in row 2, where 2 + 3 would predict five rather than the four circles actually shown. A rule must account for the known cells, not just one convenient example. D matches the required count but changes the shape without support.

Example 2: track direction and dot count independently

The arrow turns clockwise by a quarter turn at each step to the right. Test that proposal against the grid rather than assuming it is correct. The symbol “·” represents one dot.

Options: A: ↑ ·; B: ↑ ···; C: → ···; D: ↓ ···.

Answer: B, an upward arrow and three dots. In row 3, down → left → up continues the clockwise quarter turns. The other two rows follow the same rule. Down each column, the arrow also advances by one quarter turn. The dot count follows a separate relationship: one dot in column 1, two in column 2 and three in column 3.

A gets the direction right but ignores the dot count. C and D get the count right but fail the arrow rule. Splitting the features makes it possible to identify exactly why an option fails, instead of judging the whole picture at once.

If the arrow rule is clear but an asymmetric arrangement still looks like either a turn or a mirror, continue with five original rotation and reflection exercises. They add named landmarks, movement within a frame, symmetry limits and operation order.

Example 3: combine symbols, then remove the repeated symbol

In this example, each row combines its first two cells using the same operation. The order of symbols inside a cell does not matter. Compare three candidate operations: keep all symbols, keep only shared symbols, or keep only symbols appearing in exactly one of the first two cells.

Options: A: △ ☆ □; B: ☆; C: △ □; D: ○ □.

Answer: C, triangle and square, under the operation demonstrated by the completed rows. In row 1, the triangle occurs in both starting cells and disappears; the circle and square each occur once and remain. Row 2 checks the operation: the repeated circle disappears, leaving diamond and star. In row 3, remove the repeated star and retain triangle and square.

A keeps the repeated star, as an “all symbols” operation would. B keeps only the shared star. Neither operation explains the completed rows. D copies row 1's result without using row 3's inputs. This is a teaching example of one combination rule, not a claim that every matrix question uses it or that a finite grid can prove a unique rule among all imaginable rules.

What does XOR mean in a matrix explanation? XOR stands for exclusive OR: keep a symbol if it appears in exactly one input. That is the operation used above. Let A be the first cell and B the second. For each particular symbol, write 1 if it is present and 0 if it is absent; these numbers do not represent the total number of shapes.

In row 1, circle has inputs (1,0), triangle (1,1), and square (0,1). OR would retain circle, triangle and square; AND would retain triangle; XOR retains circle and square. A−B retains only circle. Reversing that subtraction, B−A, retains only square, so subtraction has a direction.

Here, presence means the identity of a symbol anywhere in a cell; its position is ignored. A different puzzle may compare matching positions or another specified feature. Identify what is being compared before using the table, and preserve any relevant differences in colour, orientation or count.

When XOR compares positions instead of symbol identities

The same logic can compare occupied positions rather than different kinds of symbol. To see the difference, use two fixed two-row, three-column boards. Every X is an identical mark and every dot is an empty position. Keep both frames aligned; do not shift or turn either board.

Input A   Input B   Position XOR
X..       .X.       XX.
.XX       .XX       ...

Compare each matching position separately:

The result is XX./...: two marks in the top row. Cancelling every X merely because the same symbol occurs somewhere in both inputs would erase the wrong information. Here the feature being compared is occupancy at each fixed position, not the identity “X” anywhere on the board. This illustrates the operation; another puzzle still needs completed cells that support using it.

Example 4: distribute shape and fill independently

Each cell contains exactly one shape. The words describe two separate features: its shape and its fill. “Open” means an unfilled outline, “solid” means a filled interior, and “striped” means an interior with visible stripes. The text labels carry all the information; no colour distinction is required.

Options: A: solid triangle; B: open circle; C: open triangle; D: striped square.

Answer: C, an open triangle. Check shape first. Each completed row contains one circle, one triangle and one square. The completed columns do too. Row 3 already contains a square and a circle, so it needs a triangle; column 3 independently gives the same prediction.

Now ignore shape and check fill. Each completed row and column contains open, solid and striped once each. Row 3 already contains solid and striped, so the missing fill is open. Column 3 contains striped and solid and confirms open. Combine the two predictions only after checking them separately: one open triangle.

A supplies the right shape but repeats solid fill in the final row and column. B supplies the right fill but repeats the circle. D supplies neither the missing shape nor the missing fill. All four options contain one object, so counting alone cannot distinguish them.

This is a distribution rule: each value appears once in the relevant row and column. It does not require one shape to turn into another or a filled interior to fade gradually. Notice that circles have different fills in different cells; “every circle is open” would contradict row 2 and row 3. As with the earlier examples, the answer follows the stated simple relationship supported by the completed cells, not a universal rule for every matrix.

What to do when none of these rules fits

Return to the features you can actually see. Check whether you missed a change in position, fill or direction, or whether two relationships operate independently. Compare a proposed rule with a completed part of the problem before applying it to the blank. If several options still seem possible, keep that uncertainty; adding an arbitrary exception is not evidence for a solution.

After practice, record one specific observation, such as “I checked the direction but missed the dot count.” A total score cannot recover that observation for you. The guide to reviewing one reasoning-test session explains how to distinguish your own notes from conclusions a score cannot support.

Turn the examples into a useful practice session

Use new material rather than repeatedly recalling the answers above. For each problem, write the proposed rule, a known cell that checks it, and the feature that rejects a tempting option. On a later problem, see whether that checking step catches an error you otherwise would have missed.

Copy a rule-check record

Copy this table into your own notes before opening an explanation. It is a plain-text record you can reuse with other practice material.

Here is a completed record for Example 3. It records the reasoning, not just the answer letter.

You can change the counts or starting directions in these examples to create a new exercise. Recalculate the full grid and the options; changing a symbol without checking the consequences can produce an ambiguous or inconsistent question. Correctly solving a self-made example is practice in applying that rule, not a standardized assessment.

For a longer routine, follow the four-week reasoning practice plan. Retest research finds that repeated administration can affect cognitive-test performance; it does not validate a particular improvement claim for this practice form. A higher score on familiar material alone does not establish an increase in general intelligence. See Scharfen, Peters and Holling's retest meta-analysis.

How this connects to the free FermatMind IQ test

The current free IQ test offers 30 original visual reasoning questions, with an estimated duration of about 20 minutes. The test and its results are free. The examples on this page are separate teaching material, so learning them does not provide a key to the test. The current form is not an official Raven assessment or a Mensa admission test.

After completing and submitting all 30 questions, you can review your correct-answer count, answer accuracy, recorded time and performance by item group. When eligible reference data is available, the result also shows a Beta reference score and a model-reference percentile with their source and version. For example, 18 correct answers give 60% accuracy; under the current designed Beta model, they map to a reference score of 126 and a model-reference percentile of 95.85%. These are different measures. The reference figures are not population IQ or rankings among actual participants. You can also save a TXT or Markdown summary. See IQ scores and interpretation limits for details.

An online practice result does not establish a clinical diagnosis, educational placement or Mensa eligibility. Mensa Norway's own practice-test instructions also distinguish its online test from professional testing; that is a statement about its tool, not evidence that FermatMind has the same norms or measurement properties. For formal assessment, ask a qualified professional which instrument and interpretation fit the purpose.

Frequently asked questions

Is a matrix reasoning test the same as a complete IQ assessment?

No. Matrix questions focus on relationships in visual patterns. A broader cognitive assessment can include other tasks and requires evidence for its intended interpretation. FermatMind's original 30-question online form is reasoning practice and does not replace professional assessment.

Are the FermatMind test and results both free?

Yes. The current original 30-question test and its results are free. Allow about 20 minutes; actual time varies. The worked examples in this article are separate teaching material, not answers to the test bank.

Should I always add the shapes or rotate them?

No. Those are possible relationships, not universal rules. Check any proposed operation against the completed cells. A rule that fails a known row should be rejected, even if an answer option looks plausible.

Do these examples reveal the answers to the real test?

No. They were created for this article to demonstrate checking a rule, separating features and rejecting an option. They are not the test bank's answer explanations and do not cover every question type.

Does a better practice score mean my IQ increased?

Not by itself. Familiarity with items and methods can affect retest performance. Record what changed in your approach and conditions, and use new material to check a strategy; a score increase alone does not establish a change in general intelligence.

Nine-example worksheet: check the rule before the option

Use this worksheet with the four examples above and the five exercises in the rotation and reflection guide. It records your reasoning; it does not create a test score.

You will describe a proposed rule, check it against information already given, and explain why an alternative fails. One exercise shows how a missing picture can be predictable even when the operation cannot be uniquely identified.

You need a browser with the linked public guides, this record and somewhere to write. Squared paper is optional. Work alone or discuss with a partner. Keep notes locally; there is no upload or collection step.

Date: ____________________

Route: Full A1–A4 and B1–B5 / Short A1, A2, B1, B4

Before you begin

The explanations are visible on the same public pages. Open the named section, inspect its instructions, diagram and options, and stop before its answer. If you see a solution early, write explanation seen and practise explaining the reasoning instead. Keep your first notes visible.

For every example, record the relevant features, state a precise rule, test it against a different completed part where another is available, predict the missing content, reject an alternative using a specific contradiction, then read the explanation and record your correction. A correct letter alone is not the learning record.

Start with the four-step method above. To take this record away, copy it into your own notes or print the relevant part of this page using your browser. Keep the exercise names with your notes so you can return to the worked explanations.

A1. Count without changing the shape

Open Example 1: count the circles.

What changes? What stays the same?

Writing space: __________________________________________________

My rule in words: _________________________________________________

Another completed row or column that checks it, with its actual counts:

Writing space: __________________________________________________

Predicted missing content: ______________ My option: ______________

Does adding the first two counts explain every completed row? Give a decisive check:

Writing space: __________________________________________________

After reading: I kept/changed my answer because ___________________

Writing space: __________________________________________________

Explanation seen before attempting? Yes / No

A2. Track direction and count independently

Open Example 2: direction and dot count.

Direction rule: ___________________________________________________

Dot-count rule: ___________________________________________________

A completed part that checks the direction rule: ____________________

Predicted direction: __________ Predicted dots: __________

My option: __________

For one wrong option, name its correct feature and incorrect feature:

Option: ______ Correct feature: ___________________________________

Incorrect feature: ________________________________________________

After reading, the check I omitted or used successfully was:

Writing space: __________________________________________________

Explanation seen before attempting? Yes / No

A3. Decide what the combination operation compares

Open Example 3: symbol combination.

This example compares symbol identities, ignoring order within a cell.

Operation supported by the completed rows: ________________________

Shared symbol in a completed row: ______ What happens to it? ______

Symbol present in only one input: ______ What happens to it? ______

Missing cell — exclude: __________ retain: __________ option: ______

Why does keeping every symbol fail a completed row?

Writing space: __________________________________________________

After reading, one condition to check before using this operation in a different puzzle:

Writing space: __________________________________________________

Explanation seen before attempting? Yes / No

A4. Check shape and fill as separate attributes

Open Example 4: distribute shape and fill independently above. Look at the diagram and options before its explanation.

Shape missing from the third row: ______ From the third column: ______

Fill missing from the third row: ______ From the third column: ______

Combined missing content: ____________________ My option: ______

A completed row AND column that verify the shape rule: ______________

A completed row AND column that verify the fill rule: _______________

Pick a wrong option and name which attribute fails: _________________

After checking, the feature I overlooked or checked correctly: _______

Explanation seen before attempting? Yes / No

Combination checkpoint: specify what is compared

Use the guide’s combination truth table and fixed-position XOR demonstration. OR keeps a feature present in either input or both; AND keeps only the overlap; XOR keeps a feature present in exactly one; left-minus-right keeps what is in the left input alone. Identify each input and its order before applying subtraction.

A3 compared symbol identities. The fixed-position demonstration compares whether each particular grid position is occupied; do not move symbols between positions before combining them.

One shared symbol removed in A3: ______ One shared occupied position removed in the fixed-position example: ______

One position retained by XOR: ______ Why would OR give a different result? ______

My input order and comparison unit in a new puzzle: __________________

This optional checkpoint supplements the nine teaching examples. No exercise in this pack produces a test score.

Read the rotation guide's notation

Open Read the diagrams before trying a rule.

The digits identify different markers. Move their positions and keep their printed labels upright; the digits are not quantities. Each small board is transformed. The outer matrix contains examples of those changes. A marker's inner board position differs from the board's outer matrix position. The guide supplies textual row codes as an alternative to the diagrams.

B1. One matching marker is insufficient

Open Exercise 1: a matching corner is not enough.

Candidate operation: ______________________________________________

Centre or axis: ___________________________________________________

Known step — marker: ______ from: ______________ to: ______________

Second marker: ______ from: ______________ to: ______________

Another completed step checking the same operation:

Writing space: __________________________________________________

Predicted missing marker positions: ________________________________

My option: ______

Alternative option/operation: ______________________________________

It matches marker: ______ but contradicts marker/position: __________

After checking, correction or retained reasoning:

Writing space: __________________________________________________

B2. Name the mirror axis at each step

Open Exercise 2: mirror axis.

First operation: __________________ Second: _______________________

Are the lines fixed to the page or carried with the markers? _________

Marker movement rejecting the two-quarter-turn schedule:

Writing space: __________________________________________________

For the missing board, start at outer cell: _________________________

Required next operation: __________________________________________

Predicted positions: __________________________ My option: ______

Why can the endpoint resemble a half-turn without each step being a rotation?

Writing space: __________________________________________________

After checking: ___________________________________________________

B3. Separate movement from turning

Open Exercise 3: placement and orientation.

The three-column footprint moves within a five-column frame.

Its starting columns across a known row: ______ → ______ → ______

Internal turning rule and its centre:

Writing space: __________________________________________________

Missing board — footprint position: ______ internal orientation: ______

Predicted frame positions — marker 1: ______ 2: ______ 3: ______

My option: ______

One wrong option: ______ Fails placement / orientation / both

A position that proves the failure: _________________________________

After checking: ___________________________________________________

B4. Say what the evidence cannot distinguish

Open Exercise 4: symmetry limits.

Operations compatible with the completed pair:

Writing space: __________________________________________________

Rejected operation and decisive contradiction:

Writing space: __________________________________________________

Predicted missing visible board: ______________ My option: ______

The missing picture is predictable because:

Writing space: __________________________________________________

The operation is not uniquely identifiable because:

Writing space: __________________________________________________

How could a newly designed labelled arrangement distinguish the surviving operations? Your addition is not part of the original evidence.

Writing space: __________________________________________________

After checking: ___________________________________________________

B5. Check operation order

Open Exercise 5: operation order.

Order 1: _____________________ Order 2: _____________________

Track marker: ______ through a known pair under both possibilities:

Order 1 — start: ______ intermediate: ______ final: ______

Order 2 — start: ______ intermediate: ______ final: ______

Which fits? ______ Another pair checking that choice: _______________

Apply it to the new arrangement. Marker 1 finishes: ______ 2: ______ 3: ______

My option: ______

Why is “a turn and a mirror” an incomplete rule?

Writing space: __________________________________________________

After checking: ___________________________________________________

The coordinate section uses board rows increasing downward. If you use another source with Cartesian y increasing upward, convert conventions before using its formulas.

Exit reflection

Which example showed why one matching feature is insufficient?

Writing space: __________________________________________________

How can one picture be predictable while several operations remain possible?

Writing space: __________________________________________________

My example of rejecting a rule using a known cell:

Writing space: __________________________________________________

One checking habit to try on unfamiliar material:

Writing space: __________________________________________________

One conclusion these exercises do not establish about me:

Writing space: __________________________________________________

Optional: attempt the new arrangement in Use the method on fresh material before reading its prediction. Record the marker movements that check your result.

Writing space: __________________________________________________

Completion

For your chosen route, check that you attempted every named example (or marked an explanation seen early), stated rules and completed-cell checks, identified a concrete contradiction for at least one alternative, distinguished output prediction from operation identification in B4, reviewed corrections and recorded one method observation without making an IQ claim.

Completion means producing and reviewing the record. Every first answer need not be correct.

Cube questions: count vertices, edges and faces separately

A cube has 8 vertices, 12 edges and 6 faces. A vertex is a corner where edges meet; an edge is a line segment where two faces meet; a face is one square surface. A flat drawing can hide some of these or make lines cross on the page, so count the structure rather than every visible mark.

The cube exercise below extends the practice to spatial counting. It is separate from the four matrix examples, the nine-example worksheet above and FermatMind’s scored test. It does not produce an IQ estimate or disclose a test-bank answer.

Build a labelled wireframe

Draw two square loops, then connect their matching corners:

Front       Back
A---B       E---F
|   |       |   |
D---C       H---G

Add the four joining edges A–E, B–F, C–G and D–H. The loops are two parallel faces of one cube, not two separate cubes. A–D are the front corners and E–H the corresponding back corners.

The complete edge list gives a text alternative to the drawing:

Read “AB” as the segment joining A and B. A crossing in a projected drawing is not an extra vertex unless the structure actually joins there.

Count in layers, then check another way

Vertices: 4 + 4 = 8. The front face has A, B, C and D. The back face adds E, F, G and H. Each label is counted once.

Edges: 4 + 4 + 4 = 12. Count the front loop, back loop and four joins. An independent check uses the three edges meeting at each vertex: 8 × 3 = 24 edge ends. Every edge has two ends, so 24 ÷ 2 = 12 edges.

Faces: 2 + 4 = 6. There are front and back faces, plus top, bottom, left and right:

Check each boundary rather than choosing any four labels. A face has four boundary edges; a diagonal across a face is not one of the cube’s edges.

A hidden corner does not disappear

Suppose a solid drawing hides H and the edges meeting there. The cube still has eight vertices. From the edge list, H joins D, E and G. Its visibility changes with the view; its place in the cube does not.

The reverse mistake is to count a line crossing as a corner. For example, if a drawing includes the diagonal AC across the front face, the diagonal adds a drawn segment but does not change the cube’s 12 edges. If both face diagonals cross, their crossing is not a new vertex of the cube.

Before counting, ask: “Am I counting the cube’s structure, visible marks, or every point in a subdivided grid?” Those are different tasks.

Subdividing a cube changes the question

Divide each original edge into two equal parts and make a regular 2 × 2 × 2 array. It contains 8 small cubes. The grid has three vertex positions along each direction, so it contains 3 × 3 × 3 = 27 distinct grid points. The original outer cube still has eight corners.

More generally, a regular n × n × n array has:

• n³ small cubes;
• (n + 1)³ distinct grid points, including interior points;
• eight corners of the original outer cube.

These formulas count different objects. A question asking for all cubes of every possible size would require an additional count; n³ answers only the number of smallest cubes.

Try five fresh checks before reading the answers

Use only the labelled cube and definitions above. Write a reason, not just a number or letter.

C1 — Neighbours. Which three vertices join H by a single cube edge?

Your answer: ____________________

C2 — Opposite corner. Which corner is opposite A across the cube: C, E, G or H? “Opposite” here means across the body of the cube, not along an edge or a face diagonal.

Your answer and reason: ____________________

C3 — Hidden vertex. A solid view shows seven of the eight labelled corners. No part of the cube has been removed. How many vertices does the cube have? Does seeing seven prove that its shape changed?

Your answer: ____________________

C4 — A new grid. A regular 3 × 3 × 3 array is drawn inside one outer cube. Give three separate counts: smallest cubes, distinct grid points, and original outer corners.

Your counts: ______ / ______ / ______

C5 — Painted outside. Paint all six exterior faces of that 3 × 3 × 3 cube, then separate its 27 small cubes. How many small cubes have paint on exactly two faces? Explain which positions they occupy.

Your answer and position check: ____________________

Answers and counterchecks

C1: D, E and G. They are joined to H by DH, HE and GH. F is tempting because it is on the back face, but HF would be a face diagonal.

C2: G. A and G differ in all three front/back, left/right and top/bottom positions. C is across the front face; E is joined directly to A; H lies across the left face. None of those is the body-opposite corner.

C3: Eight. The condition says it remains a complete cube. A hidden corner changes what the view reveals, not the number of vertices. If the object had actually been cut, the cube assumption would need to be checked again.

C4: 27 / 64 / 8. The smallest cubes number 3³. Grid points have four positions in each of three directions, giving 4³. Only eight of those are corners of the original outer cube.

C5: Twelve. Each of the 12 outer edges contains one middle small cube with exactly two painted faces. Exclude the eight corner pieces, which each have three painted faces. As a complete check, the 27 pieces split into 8 with three painted faces, 12 with two, 6 face-centre pieces with one, and 1 interior piece with none: 8 + 12 + 6 + 1 = 27.

Copy this short record for another spatial problem:

Object being counted: ____. Labels or layers used: ____. First count: ____. Independent check: ____. Tempting wrong count: ____. The feature that disproves it: ____.

To continue with two-dimensional transformations, use the rotation and reflection exercises. If you later take the original 30-question reasoning exercise, keep this worksheet as practice notes rather than treating it as a score conversion or an answer key.

How to create an original reasoning exercise

You can create a small reasoning exercise by choosing a learning goal, writing an exact rule, generating every cell and checking the alternatives. Here is a complete original example you can reproduce on paper. It is teaching material created for this guide, separate from FermatMind’s scored questions.

1. Choose one observable learning goal

The goal is to practise checking quantity and direction independently. A successful explanation must state what happens to the count, what happens to the arrow and which known cells check those relationships. Choosing the correct letter alone does not complete that explanation.

Each cell contains identical dots and one arrow. To keep the grid readable, the tables use a number for the exact dot count: 3, → means three dots and one right-pointing arrow. The number is a quantity here, not a marker’s name. Dot positions, spacing, size and fill carry no additional information.

2. Write the design rules before drawing

Use these rules throughout this exercise:

• Moving one cell right adds two dots and turns the arrow 90 degrees clockwise.
• Moving one cell down adds one dot and turns the arrow 90 degrees counterclockwise.
• The arrow cycle clockwise is up → right → down → left → up. Counts do not wrap around.

Start the top-left cell with one dot and an upward arrow. Applying either route to an interior cell must give the same result. For example, moving right and then down adds three dots overall and returns the arrow to its starting direction.

These are the designer’s explicit rules. You can show them when teaching rule application or keep them beside your own answer sheet when asking someone to infer the relationships.

3. Generate the complete grid first

Do not start by drawing a blank and guessing its answer. First calculate all nine cells:

Check every cell against the starting point:

Also check adjacent cells directly. Every horizontal step must satisfy both horizontal rules, and every vertical step must satisfy both vertical rules. This catches a copied or mistyped cell that a single final-answer check could miss.

4. Remove a cell and write the question

Now replace only the bottom-right cell with a question mark:

Use this instruction:

Counts change by a fixed amount across each row and by a fixed amount down each column. Arrows make a consistent quarter-turn at each horizontal step and a consistent quarter-turn at each vertical step. Find the missing count and direction. Check both a completed row and a completed column.

The prompt identifies the intended family of relationships without supplying the amounts or turn directions.

5. Give each wrong option a specific purpose

A distractor is a wrong answer designed around a possible mistake. Make the options distinct and explain the intended mistake in your answer sheet:

These are hypothesised mistakes for teaching. We have not measured how frequently learners make them.

Answer: C, seven dots and an upward arrow. From the cell to its left, five dots become seven and left turns clockwise to up. From the cell above, six dots become seven and right turns counterclockwise to up. The two routes agree.

A fails the horizontal count rule. B fails the horizontal turn. D uses 3 + 5, but adding the first two counts does not explain the completed first row: 1 + 3 would give four, whereas five is shown. E reverses the horizontal turn; the completed first row shows up → right → down.

6. Check ambiguity before sharing

Confirm that the legend defines every meaningful feature, every known cell follows the rules, and exactly one listed option satisfies both required attributes. Remove duplicate options, including differently worded options that describe the same cell.

Then challenge the intended explanation. Does a simpler competing rule fit the known cells and predict another option? If so, revise the prompt, grid or options and recalculate everything.

For this exercise, C is the only listed option that satisfies the specified count-and-quarter-turn rules. That is a limited, checkable conclusion. A finite grid cannot rule out every imaginable alternative pattern, so do not describe it as proving one uniquely possible rule among all rules.

7. Try the same generator from a second starting point

Change the top-left cell to two dots and a right-pointing arrow. Keep all four transformations unchanged, regenerate the grid and hide the final cell:

Answer: eight dots and a right-pointing arrow. Moving right from 6, ↑ adds two and turns clockwise. Moving down from 7, ↓ adds one and turns counterclockwise. Both produce 8, →.

This version supplies a useful countercheck: adding the first two counts happens to work in row 1, because 2 + 4 = 6. It fails in row 2, because 3 + 5 is eight, not the seven shown. One matching row is still insufficient.

The second starting point checks the generator’s consistency; it is not an independent assessment of someone who has already learned the rule. If another reader tries your exercise, ask them to explain the legend, rule and contradictions. Unexpected interpretations can reveal wording problems, but a successful reader check does not establish measurement validity.

What would developing an IQ assessment require?

A working puzzle is one educational item. Developing an assessment with interpretable IQ scores requires evidence beyond checking its answer.

The Standards for Educational and Psychological Testing address:

• defined constructs, intended uses and populations;
• test specifications, item development, review, tryouts and item analysis;
• score precision and evidence supporting each intended interpretation;
• fairness and barriers unrelated to the intended construct;
• documented reference populations and norming methods, where norms are used;
• reproducible administration and scoring procedures.

This is a development outline, not a recipe that guarantees validity. No universal sample size or homemade IQ conversion follows from this exercise. Keep its output as a rule explanation and learning record. For the distinction between raw performance, norms and standard scores, read what IQ is and how it is measured.