Aptitude skill lesson
Inductive reasoning: skill lesson
Inductive reasoning is the ability to look at a handful of examples (a number series, a set of observations, a table of past decisions), and propose the rule that generated them, while staying honest about the fact that finished evidence never fully proves a rule. It carries most of the weight in graduate ability batteries, analyst and intelligence screening, and college placement testing, and it is what a technician does when three failures on the same line suggest a pattern rather than three coincidences. The construct rewards two opposite habits at once: generate candidate rules quickly, then try hardest to break the one you like. Everything you practise here is device-local unless you export it.
Published · last reviewed
Applies to Novus Learn 0.1.0
What changed, and when
- , Replaced the body of all 27 non-judgement lessons with construct-specific material: six worked examples each, carrying the actual arithmetic, the actual inference or the actual procedure, plus expanded objectives, practice tips and glossary. Every 'Related Learn topics' link now points at a real page on this site rather than a generic search. The eight workplace-judgement lessons are unchanged.
- , Rebuilt the lesson page around a sticky contents rail, per-section links, previous/next lesson navigation, and two graded checkpoints drawn from the open practice bank.
- , Repaired the aptitude integrations behind the lessons so each one links to skill-specific practice instead of the unscoped fixture engine.
- , Published one lesson for each of the 35 aptitude skill constructs: objectives, worked examples, practice tips, glossary, Learn topic links, and sources.
These 35 skill lessons are authored and revised as one set, so they share one revision history rather than 35 identical dates.
Objectives
Copy link- Compute first and second differences on a numeric series and use them to distinguish arithmetic, quadratic and multiplicative rules.
- Detect interleaved series, where alternate terms follow two separate rules, before concluding a sequence has no pattern.
- Convert letter sequences to alphabet positions, apply the same difference machinery, and wrap correctly past position 26.
- Explain why a finite sequence is under-determined, and choose between competing rules by testing each against every given term rather than the first two.
- Induce a decision rule from a small table of labelled cases, and identify the single row that rules out the obvious over-general hypothesis.
- Design the test that could disconfirm your hypothesis, and recognise why confirming cases are worth far less than disconfirming ones.
Checkpoint: does the idea land?
Copy linkBefore the worked examples, check that the objectives above actually landed.
Two questions from the open practice bank, answered here and scored on this device. Untimed, ungraded, and not added to your practice history, the full bank is where attempts are recorded.
Examples
Copy linkDifferences first, ratios second
Four series, each solved by the same opening move. Series one: 2, 6, 12, 20, 30. First differences are 4, 6, 8, 10: arithmetic, rising by 2, so the next difference is 12 and the next term is 42. The closed form is n times n-plus-one: 1x2, 2x3, 3x4, 4x5, 5x6, 6x7 = 42, which confirms it. Series two: 7, 10, 16, 28, 52. First differences are 3, 6, 12, 24. Those are doubling, so the next difference is 48 and the next term is 100. Equivalently each term is double the previous minus 4: 7x2-4 = 10, 10x2-4 = 16, 16x2-4 = 28, 28x2-4 = 52, 52x2-4 = 100. Series three: 3, 4, 7, 11, 18, 29. Nothing simple in the differences, so try summing neighbours: 3+4 = 7, 4+7 = 11, 7+11 = 18, 11+18 = 29, and the next term is 18+29 = 47. Series four: 1, 4, 9, 61, 52. That looks broken until you notice 61 and 52 are 16 and 25 with their digits reversed; the underlying series is the squares 1, 4, 9, 16, 25, 36, written back-to-front where reversing changes anything, so the next term is 63. The order of attack that solves most series is fixed: first differences, then second differences, then ratios, then sums of neighbours, then digit manipulation. Run it in that order and you stop staring.
Two rules fit; the fourth term decides
You are shown 2, 4, 8 and asked for the next term. Doubling gives 16. But first differences are 2 and 4, and if those differences are themselves rising by 2 the next difference is 6 and the next term is 14. Both rules fit every term you were given, so the sequence is genuinely under-determined and neither answer is more correct than the other from the data alone. This is not a trick; it is the central limitation of induction, and assessment writers manage it by giving you enough terms to separate the candidates. Add one term. If the series is 2, 4, 8, 14 then the differences are 2, 4, 6 and the next difference is 8, giving 22, doubling is dead, because doubling would have produced 16 at position four. If the series is 2, 4, 8, 16 the difference rule is dead instead, because it predicted 14. So the practical rule is: never commit on two terms, always test your candidate rule against the LAST given term as well as the first, and if two rules survive all given terms, look at the answer options, exactly one of them will normally correspond to a rule that fits, and that is legitimate evidence about what the writer intended.
Letters are numbers wearing a costume
The sequence C, F, J, O, U. Convert to alphabet positions: C is 3, F is 6, J is 10, O is 15, U is 21. First differences are 3, 4, 5, 6, rising by one, so the next gap is 7 and the next position is 28. The alphabet has only 26 letters, so wrap: 28 minus 26 is 2, which is B. Answer B. Three things go wrong on letter items. Candidates count gaps by reciting the alphabet on their fingers and drop a letter, which is why writing the numbers down beats counting in your head every time. Candidates forget to wrap, and answer with a position number rather than a letter. And candidates mishandle a series that runs backwards past A. Position 1 minus 3 is minus 2, which wraps to 26 minus 2, that is 24, the letter X. A related family uses letter pairs where the two letters move at different rates, for example AZ, CX, EV, GT: the first letters are 1, 3, 5, 7 going up by two and the second are 26, 24, 22, 20 going down by two, so the next pair is I and R, that is IR. Split the pair, treat each stream separately, and the item becomes two easy arithmetic series instead of one impossible one.
Interleaving: two clocks in one series
The series 5, 8, 6, 11, 7, 14, 8. First differences are 3, minus 2, 5, minus 4, 7, minus 6: alternating sign, no obvious progression, and this is the point where candidates guess. Split it instead. Terms in the odd positions are 5, 6, 7, 8, rising by one. Terms in the even positions are 8, 11, 14, rising by three. The next term sits in an even position, so it continues the second stream: 14 plus 3 is 17. The diagnostic that should trigger the split is alternating signs in the first differences, or a series that oscillates while drifting. Interleaving also appears with three streams, and with one stream constant: 4, 9, 4, 16, 4, 25 hides the squares 9, 16, 25 among repeated 4s, so the next term is 4 and the one after is 36. One caution worth carrying: a series that has been split should be checked back against the original by writing out your predicted term in place and reading the whole thing again. If the reassembled series looks stranger than the one you started with, the split was probably wrong.
Inducing a rule from labelled cases
Five support tickets, each with a customer tier, a first-response time, and whether it was escalated. Ticket 1: Enterprise, 6 hours, escalated. Ticket 2: Enterprise, 2 hours, not escalated. Ticket 3: Standard, 9 hours, not escalated. Ticket 4: Enterprise, 5 hours, escalated. Ticket 5: Standard, 1 hour, not escalated. Three hypotheses are worth writing down. Hypothesis A: escalation happens when the response is slow. Ticket 3 kills it. A nine-hour Standard ticket was not escalated. Hypothesis B: escalation happens for Enterprise customers. Ticket 2 kills it. A fast Enterprise ticket was not escalated. Hypothesis C: escalation requires BOTH Enterprise tier and a response over four hours. It fits all five rows, and it is the only conjunction that does. Notice which rows did the work: the two that fit one hypothesis but not the label are worth more than the three that agree with everything. Now the honest limit. Does an Enterprise ticket answered in exactly four hours escalate? Hypothesis C as written says no, but the data contains no case between two and five hours, so the true threshold could be anywhere in that window. Induction from five rows gives you a rule and a region of ignorance, and naming the region is part of the answer.
Look for the case that would break you
Four cards lie on a table showing A, K, 4 and 7. Each card has a letter on one side and a number on the other. The rule to test: if a card has a vowel on one face, it has an even number on the other. Which cards must you turn? Most people say A, and many add 4. A is right: a vowel with an odd number on the back refutes the rule directly. The 4 is useless: the rule says nothing about what must be behind an even number, so a consonant there breaks nothing and a vowel there merely agrees. The card people miss is the 7. Turn it, find a vowel, and the rule is dead. So the answer is A and 7, the two cards that could produce a violation. This is the Wason selection task, a well-known public result in the psychology of reasoning, and the reason it belongs in an inductive lesson is that it isolates the habit the construct actually rewards: people search for confirming evidence by default and have to be trained to search for disconfirming evidence. Carry it into series items as a check. Once you have a candidate rule, do not run it forward on the terms it was built from; run it on the term you have not used yet, and treat a single mismatch as fatal rather than as noise.
Checkpoint: can you apply it?
Copy linkNow apply it. These come from a later section of the bank, so they are not more of the same.
Two questions from the open practice bank, answered here and scored on this device. Untimed, ungraded, and not added to your practice history, the full bank is where attempts are recorded.
Practice tips
Copy link- Write the first differences under every numeric series before you think about it at all. The mechanical step surfaces arithmetic and quadratic rules for free and tells you within seconds whether to move on to ratios.
- Fix an order of attack and follow it: first differences, second differences, ratios, sums of adjacent terms, then digit tricks. Most lost time in this construct is spent re-staring at a series rather than working through the list.
- Convert letters to numbers on paper the moment you see a letter series, and write the alphabet with positions at the top of your rough sheet once at the start of a session rather than counting on your fingers per item.
- Treat alternating signs in the first differences as a direct instruction to split the series into odd and even positions. The interleaved family is large and is where candidates most often decide, wrongly, that there is no pattern.
- Validate a candidate rule against the last given term, not the first two. A rule that explains the opening of a series and misses its final term is the standard wrong answer, and the answer options are usually built to reward it.
- When you induce a rule from a table of cases, name the row that eliminated the simpler hypothesis and name the gap the data leaves open. Both are recorded on this device only, so build the habit here where over-claiming is free.
Glossary
Copy link- Inductive inference
- Reasoning from particular observations to a general rule. The conclusion is supported but never guaranteed, which is the essential difference from deduction.
- First difference
- The gap between consecutive terms of a series. A constant first difference means an arithmetic progression; a constant second difference means a quadratic rule.
- Geometric progression
- A series where each term is the previous term multiplied by a fixed ratio. Detected by dividing neighbours rather than subtracting them.
- Interleaved series
- A single printed sequence containing two or more independent series in alternate positions. Signalled by first differences that alternate in sign.
- Under-determination
- The condition in which several different rules fit every term you were given, so the data alone cannot single one out. It is the reason short series need extra terms or answer options to be decidable.
- Disconfirming instance
- A case that a hypothesis predicts should not exist. One of these refutes a rule outright, whereas any number of agreeing cases only fails to refute it.
- Confirmation bias
- The tendency to seek evidence that agrees with a hypothesis already held. In series and rule-induction items it shows up as checking a rule against the terms that suggested it.
- Wrap-around
- Continuing a letter sequence past Z back to A, or before A back to Z, by adding or subtracting 26 from the alphabet position. The routine source of near-miss answers on letter items.
Sources
Copy link- Worked series written for Novus Learn from standard sequence analysis: first and second differences, geometric ratios, neighbour sums, alphabet-position arithmetic, and interleaved streams. Every series above was extended and checked term by term.
- The four-card selection problem is the Wason selection task, first published by P. C. Wason in 1968 and widely reproduced in the public psychology-of-reasoning literature; the description above is a plain restatement of the standard version, not an item from any commercial test.
- Terminology checked against the public Wikipedia articles 'Inductive reasoning', 'Arithmetic progression', 'Geometric progression' and 'Wason selection task'. Definitions only.
- Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
- Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.
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