The Series That Looks Like Pairs: 2, 3, 6, 7, 14, 15, ?
Dev, a ninth grader, reviews the 20-item number-series section he finished in 11 of its 14 minutes; he scored 14/20. He keeps a log of answers and time per item. Item 9 read 2, 3, 6, 7, 14, 15, __ and he wrote 16 in eight seconds. The key says 30, and the key lists only answers, not rules, so Dev cannot see what went wrong - he says the sequence 'obviously goes up in pairs: 2 and 3, then 6 and 7, then 14 and 15, so the next one is the second of a pair.' He got item 18 right (3, 4, 7, 11, 18, 29, __ = 47) by adding, and item 13 right in four seconds (64, 32, 16, 8, __ = 4). His log shows an average of about 20 seconds on items he gets right and about 45 on items he gets wrong.
1. Explain why 30 is correct and why Dev's 16 is not. What test separates a real rule from a description of the numbers?
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Dev's 'pairs' observation is true but it is not a rule. It describes the shape of the list without predicting anything: it never says how a pair is chosen or how one pair leads to the next. It also leaves the two jumps unexplained - 3 to 6 and 7 to 14 - and those jumps are where the pattern actually lives.
30 comes from one rule applied at every step: +1, then ×2, alternating. 2 + 1 = 3, 3 × 2 = 6, 6 + 1 = 7, 7 × 2 = 14, 14 + 1 = 15, 15 × 2 = 30. Six transitions, no leftovers.
The separating test has two parts. (a) The rule must generate every printed term, not just the last few. (b) It must be the same operation repeated, not a fresh description invented at the end. Dev scores zero on (a), explaining at most three transitions by hand-waving, and his rule cannot be run backward at all.
The repeated pause-then-leap is also the visible signature of two interleaved rules, which is the cue he should have read instead of stopping at 'pairs.'
2. Give Dev a two-question check he can run in under 20 seconds on any series, and show it on item 9.
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Q1 - Where does the rhythm change? Scan the gaps: +1, +3, +1, +7, +1. A small step followed by a large step, repeatedly, means two rules alternating. Split by position and read each strand alone: odd positions 2, 6, 14, __ (×2, then +2 each time) gives 30; even positions 3, 7, 15 also follow a ×2 + 1 step. Both strands are consistent, which confirms 30 independently of the +1/×2 reading.
Q2 - Can I walk it backward? Start from the answer and apply inverse operations until you reach a printed term. 30 ÷ 2 = 15 ✓, 15 − 1 = 14 ✓, 14 ÷ 2 = 7 ✓. If the chain cannot be walked back, the answer is a guess.
Why this catches Dev's error: 16 survives Q2 only in the sense that '15, 16' is a plausible-looking pair, but it fails Q1, because it leaves the jumps unexplained and cannot produce 14 from 7. Cost on a seven-term series is about 15 seconds.
3. Dev finishes early but burns 45 seconds on the items he misses. What does that pattern suggest, and what should he change about pacing?
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The pattern is the signature of being stuck between two candidates with no tiebreak, not of carelessness. On items he solves, the rule arrives in about 20 seconds; on items he misses, he spends more than twice as long, which is exactly what cycling between two unverified guesses looks like. He also leaves three minutes unused, so 'slow down and be careful' is the wrong prescription - he is already slow where slowness hurts and fast where it does not.
The change is a decision point rather than a speed target. At roughly 30 seconds, if no rule has been confirmed by the two-question check, he should mark a best guess and move on. The banked three minutes then go to the flagged items, where a fresh look plus the check is worth more than another 15 seconds of staring.
Second change: pre-scan the gaps before computing anything. A rhythm change in the first five seconds tells him to expect two interleaved rules, so the hard items stop being surprises.