Solve numeric sequence pattern puzzles

Optiver

Aug 13, 2025

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Sequence Pattern Puzzles

This is a timed pattern-recognition section of the kind used on quantitative-trading aptitude tests. You are given five independent integer sequences. For each one, determine the next term and state the single rule you used to get it.

Assume each sequence is governed by one consistent rule that reproduces every given term (not just the last gap). Scoring is +1 for a correct next term, −1 for an incorrect one, and you have roughly one minute per sequence.

$20,\ 40,\ 50,\ 110,\ 115,\ 215,\ ?$
$15,\ 35,\ 45,\ 105,\ 110,\ 210,\ ?$
$4,\ 3,\ 7,\ 9,\ 10,\ 27,\ ?$
$3,\ 7,\ 13,\ 19,\ 29,\ 37,\ ?$
$19,\ 18,\ 20,\ 60,\ 15,\ 14,\ 16,\ ?$

Constraints & Assumptions

Each sequence has exactly one intended rule; the five rules are unrelated to one another.
The rule must reproduce all supplied terms, and must yield one unambiguous next term.
All given terms are integers, and the intended next term is an integer.
No external lookup is available; you must reason from the digits alone under time pressure.
A blind guess has negative expected value (−1 vs +1), so only commit when a rule fits every term.

Clarifying Questions to Ask Guidance

Is each sequence independent, or do they share a theme (e.g. is sequence 2 a transform of sequence 1)?
Must the rule reproduce every term exactly, or only predict the next gap?
Is the next term guaranteed to be an integer (ruling out fractional extrapolations)?
Position matters for interleaved sequences — should I count the "?" position to decide which subsequence it continues?
Given the +1/−1 scoring, is partial credit available for stating a plausible-but-wrong rule, or is it strictly the numeric answer that scores?

What a Strong Answer Covers Guidance

A strong answer is judged on the method and justification, not just the five numbers. The interviewer looks for:

A systematic toolkit , applied in a sensible order: first/second differences → interleaving → operation cycles → number-theoretic structure (primes, powers, digits). The candidate shouldn't appear to guess.
Reproduction of every given term , with at least one internal cross-check where possible (e.g. two interleaved subsequences that agree on the same acceleration, or one sequence shown to be a termwise transform of another).
Parsimony / Occam's razor : when more than one rule fits the given terms, choosing the rule with the shortest description and a recognizable structure, and being able to articulate why it beats a more contrived alternative.
Correct handling of the "?" position in interleaved sequences (which subsequence does it continue?).
Calibrated commitment under the +1/−1 scoring: confidence tied to whether the rule is exact (e.g. primality) versus extrapolated.

Follow-up Questions Guidance

For sequence 4, contrast "every other prime" with an interleaved-difference reading and justify which a grader should accept. How does description length decide it?
Any finite sequence admits infinitely many rules. How do you formalize "simplest rule" (e.g. Kolmogorov complexity / minimum description length), and how does that principle guide your choice here?
If a sequence had two equally simple rules giving different next terms, what would you report given the −1 penalty?
For sequence 5, what makes an operation-cycle rule more or less convincing — and why does the cycle landing on clean integers (the $\div 4$ step) strengthen it?

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