From cases to a rule

Inductive reasoning test scored on how long the rule took to find

Sixteen rows of five figures, each asking what comes sixth, free and with no clock on any of them. The rows are not independent: they arrive in four unmarked runs of four that share one generating rule, so the page can report the time your first solution in each run cost against the times of the ones after it. That difference is what going from particular cases to a general rule actually costs, and a count of right answers hides it.

100% free
No signup
16 rows, 4 rule families
Discovery time vs application time
6 choices a row

16 rows of five figures, each asking what comes sixth. No clock, six choices an item, and one thing about the design you should know before you start: the items are not independent. They arrive in runs that share a generating rule, the run boundaries are not marked, and what this page reports is how long the first solution in each run took against the ones after it.

A worked one, using a rule that is not in the set

The square steps two positions along the grid at every figure and wraps round the end, so the sixth is at the position after the last one. Nothing else moves.

This example uses a position rule on purpose: none of the four rules in the scored set is a position rule, so working through it hands you nothing that the measurement depends on. That is not caution for its own sake — a warm-up drawn from the scored families would give away one of the four discoveries the page exists to time.

The properties that can carry a rule are the ones you can see: which shape, how it is shaded, how many there are, and which way it points. Knowing the vocabulary is fair and is what a real inductive test assumes; knowing which of them is in play on this particular row is the work.

Answer with a click or the number keys 1 to 6. Take as long as you like on any item — the time is the measurement, not a limit.

How to read a discovery premium

One headline figure, four family panels, and one thing to check before believing either.

Answer the rows without trying to spot the boundaries
The four families run in blocks of four and nothing marks where one ends. Hunting for the seam is wasted effort and will distort your own numbers, because the measurement is what your first solution inside a block cost, not whether you noticed a block started. The shapes and shading are redrawn on every row precisely so that a family cannot be recognized by how it looks — only by how it behaves.
Read the headline as a difference, not a speed
The figure at the top is the mean time on the rows where you first cracked a family minus the mean time on the rows you solved afterwards inside the same family, with a 95% Welch interval on it. Because both sides carry your reading speed, your mouse and your mood, those cancel; what is left is the search. An interval that straddles zero is the page telling you this run cannot separate the two, which is a real outcome and not a failure.
Then look at which item in each family you first got right
Each family panel shows its four rows as time bars, the first correct one in amber and the later correct ones in green. First-at-item-one across all four families with short bars means the rules were visible on sight and the premium is small for a good reason. First-at-item-three with a long bar and short bars after it is the classic induction shape. A family with no amber bar at all is a rule you never found, which the panel says plainly rather than folding into an accuracy figure.

Technical specifications

Rows and structure	16 scored rows in 4 families of 4, families in a random order, boundaries unmarked. A family's rule parameters are pinned — the same turn per step, the same property cycling — while the shapes, shading and starting state are redrawn on every row
The four rules	A constant turn each step; a constant growth in element count; a three-step cycle on one property; and two clocks at once, where a property alternating every step runs against a property cycling every three, so the pair only repeats after six
Answer choices	6 per row. A blind guess is worth 0.167 of a row and 16 blind guesses average 2.7 correct, which is the figure your score is placed against rather than any population
What is timed	Every row, from the frame that painted it to the timestamp on your answer. Nothing is cut off and no row has a limit; the duration is the result rather than a constraint on it
Feedback during the run	None, deliberately. A verdict after each row would turn every later row in a family into a confirmation rather than a discovery, and the quantity being measured would stop existing
How the premium is computed	Mean of the first correct row in each family, minus the mean of every later correct row, with a Welch interval at 95% — unequal variances and unequal group sizes, because first solutions are far more variable than applications
Warm-up example	One fully worked row on the start screen, using a position rule that appears in none of the four families, so working through it gives away no discovery the run is about to measure
What leaves the page	Nothing. Row durations live in an array in this tab; the copy button writes the family table and two means to your clipboard, and no per-row answer is in it

Frequently asked questions

What makes a test inductive rather than deductive?

Induction goes from cases to a rule and deduction goes from a rule to a case. A figural series hands you five instances of something and asks you to state the sixth, which requires recovering the generating rule from examples — and the rule is never given, never confirmed, and might be one of several that fit the five you can see. Deduction is the other direction: you are given premises and asked what must follow, and there is nothing to discover. That is why the two produce different-looking tests even when the material is identical.

Why does the page put four items with the same rule together?

Because otherwise the two halves of induction cannot be separated. If every row uses a fresh rule, then every row is a discovery and the run reports one blended number covering search and application at once. Grouping four rows under one pinned rule means the second, third and fourth are applications of something already found, so subtracting their times from the first one leaves the search. Most online series tests shuffle their rules and then report accuracy, which is a mixture presented as a measurement.

Why is there no feedback after each row?

Because telling you that row seven was right would collapse the measurement on rows eight, nine and ten. Once a solution has been confirmed, the later rows of a family stop being applications of a hypothesis and become checks of a known fact, and their times drop for a reason that has nothing to do with your reasoning. The cost is that a wrong hypothesis can persist for four rows, which is a real loss and shows up in the panel as a family with no first solution.

What does it mean if I never found one of the four rules?

It means that family has no discovery time and contributes nothing to the headline, and the panel says so instead of scoring it as four wrong answers and moving on. Two rules are worth knowing about here: the two-clocks family is much harder than the other three because two properties advance at different periods and each one alone looks broken, and the growth family is much easier because a count is the one property people check first. A miss on the hard one and a hit on the easy ones is the ordinary pattern.

Does a small discovery premium mean I learn quickly?

It means these four rules were quick for you, which is a narrower claim than it sounds. A premium near zero has two very different causes: the rules were obvious on sight, so there was nothing to search; or you never found them and the correct answers were lucky. The family panels separate those — the first case shows amber bars at item one with short times, the second shows few correct rows anywhere. Nothing here has been checked against how fast anybody learns anything outside this page.

Why six answer choices rather than eight, like the matrix pages?

Because a series has fewer degrees of freedom to build a plausible wrong answer from. A matrix cell can be wrong in shape, shading, count, size, orientation, an extra mark or which positions are occupied, which comfortably supports seven distractors that each look reasonable. A single figure at the end of a row supports about five before the extras become obviously silly and start giving the answer away by elimination. Six choices with five defensible distractors is a better item than eight with three throwaways.

Can I practice my way to a shorter discovery time?

You can get quicker at these four rules, which is the least interesting version of the question and the only one this page can answer. The families are drawn from a fixed set, so a second visit meets the same four in a different order with different parameters, and the second premium will be smaller because you now know what kinds of rule are on the menu. That is exactly the contamination that makes a repeated cognitive test hard to interpret, and it is why the number to compare is a first run against a first run.

Why induction has two costs, and what a single accuracy figure does to them

Figural series are older than matrices in the testing literature and they were built for a narrower question. Thurstone’s factor work in the 1930s separated a reasoning-and-induction factor from the verbal, numerical and spatial ones, and the series item — five instances, state the sixth — became the standard way of getting at it, because it removes vocabulary and arithmetic from the path between the person and the rule. What it does not remove, and what nobody noticed for a long time, is that the item measures two different operations glued together. Finding the rule is a search through a space of candidate rules. Applying it once found is a lookup. A test that counts right answers adds a search cost and a lookup cost together and reports the total, and two people with identical totals can have arrived at them in opposite ways.

Separating them needs the one design feature this page has and most online series tests do not: consecutive items generated by the same rule with its parameters pinned. Once four rows in a row are the same rule wearing different shapes, the first row you solve is a search and the rest are lookups, and the difference between those two durations is available without any norm, any comparison group or any assumption about anybody else. It is also robust in the way differences usually are — your reading speed, your monitor, your pointer and your alertness are in both sides of the subtraction and cancel out of it. What is left is a number about you, measured against you.

There is no population figure to put beside it, and it is worth being exact about why. A discovery time depends entirely on which rules the page chose and how many candidate rules a visitor believes are on the menu — change either and the number moves, so an average quoted from anywhere would describe a test rather than people. What this page reports instead is a within-visitor difference with a Welch interval on it. If you want the same figural material scored on accuracy against a 3x3 grid, the matrices page does that, and the diagrammatic reasoning test asks the same discovery question about operators rather than sequences. A run that measured how you prefer to receive information rather than how you extract a rule from it would be the VARK questionnaire, which is self-report and is a different kind of thing entirely, and the sibling that catches the answer you reached without searching at all is the cognitive reflection test.

A reaction time here is the interval between the frame that painted the stimulus and the timestamp the browser attached to your key, both read from the same monotonic clock. What neither can see is the display pipeline behind it, so on a 60 Hz screen roughly 16 ms of every figure below is the machine rather than you. That is the timing floor: two numbers closer together than that are the same number, and this page reports no precision it cannot support.

Both sides of the subtraction carry that floor and it cancels, which is the structural reason a difference is worth more than a level on a browser page. The absolute row times printed in the family panels do carry it, and they are shown to a tenth of a second for that reason rather than to the millisecond.

This is a measurement exercise, not a clinical assessment. It reports what you did on this page against a stated reference and nothing more — it cannot establish a learning disability, a processing-speed deficit or how quickly you pick things up in general. Only a qualified professional, working with more than a browser, can make that judgment.

Where the sixteen durations are held

Every number on this page is worked out by JavaScript running in the tab you are reading it in. Your answers, your reaction times and your score are never uploaded, logged or kept — which is also why the test carries on working after you disconnect from the network, and why nothing here can be held back behind an email address.

The family panels are drawn from an array of sixteen durations and sixteen answers held in this tab for as long as the results are on screen. There is no upload, no analytics event carrying a time, and no way for a second visit to know what your first one did — which is also why comparing two runs means keeping the copied summary yourself.