I’m practicing number sequence puzzles for fun, and I keep running into this brain-itch: sometimes multiple patterns seem to fit the first few terms, and I can’t tell which one is the “right” one. For example, with a sequence like 7, 10, 16, 28, 52, ?, I can spot more than one plausible rule that matches the early terms but suggests different next numbers. I get excited and latch onto the first pattern I notice, and then a later term breaks it. Rinse, repeat.
What’s a practical way to reason through this without overfitting? Is there a go-to checklist you use (like checking differences, ratios, alternating steps, parity, index-based formulas, digit patterns, etc.) and a sensible order to try them? How many terms do you typically need before you trust a pattern? And when do you decide there just isn’t enough information to pick a unique rule?
I’m looking for strategies to avoid false positives and a smarter way to test competing hypotheses quickly.
Ready to make maths more enjoyable, accessible, and fun? Join a friendly community where you can explore puzzles, ask questions, track your progress, and learn at your own pace.
By becoming a member, you unlock:
3 Responses
You’re right to notice that many early-term fits are possible-given n terms you can always invent a degree-(n−1) polynomial that matches them all-so I try to bias myself toward the simplest rule that explains every term I’ve seen. My quick checklist goes in this order: check first differences (arithmetic trend); then second differences (quadratic); then ratios (geometric); look for multiplicative behavior in the differences (e.g., differences doubling); test a few tiny linear recurrences with small integers like a_{n}=a_{n−1}+a_{n−2}, a_{n}=2a_{n−1}+c, or a_{n}=a_{n−1}+kn+c; check for alternating patterns (even/odd indices behaving differently); and finally consider index-based pieces such as n, n^2, 2^n added to a simple baseline. Two practical habits help avoid overfitting: hold one term “in reserve” as a prediction test, and prefer the rule you can describe in a short sentence with small integers. If two equally simple rules survive and predict different next terms, I’d say the data are underdetermined-ask for one more term. A friendly overview of these ideas (especially finite differences) is here: https://www.mathsisfun.com/numbers/number-sequence.html
Quick example with your sequence 7, 10, 16, 28, 52, ?. First differences are 3, 6, 12, 24; those double each step, so I’d predict the next difference is 48, giving 52 + 48 = 100. A cross-check via a tiny recurrence also works: notice each term doubles then subtracts 4 (10 = 2·7−4, 16 = 2·10−4, …), so a_{n+1} = 2a_n − 4, which again predicts 2·52−4 = 100. With only the first four terms, many other rules would still be viable (e.g., some cubic), so I’d want at least one extra term to discriminate. Rule of thumb: you need at least one more confirmation than the minimum required to “solve” a suspected model (e.g., after constant second differences appear twice, see them a third time), and if competing simple hypotheses disagree on the next term, you don’t have enough information yet.
I’d use a quick checklist in this order: first and second differences; if growth looks multiplicative then ratios/logs; check alternating steps, parity/digit cycles, and short recurrences or low‑degree polynomials, then pick the simplest rule (fewest parameters/shortest description) that fits all terms. Roughly, trust a rule only when you have at least one extra term beyond its parameters (e.g., 3 for a line, 4 for a quadratic), and if two equally simple rules still fit-your 7,10,16,28,52 has doubling differences so 100 is natural, but other stories exist-treat it as underdetermined and get another term.
When several rules fit early terms, I use a simple triage: check first differences, then second differences, then ratios; if nothing clean appears, look for alternating strands or a short linear recurrence with small integer coefficients, and only then consider index-based or digit patterns. For 7, 10, 16, 28, 52 the differences are 3, 6, 12, 24, which double, so I’d expect the next difference 48 and the next term 100. The same data also satisfy a_{n+1} = 2a_n − 4, which implies those differences double, so these two views agree and that boosts confidence. As a rule of thumb, I ask for one more confirmation than the rule’s complexity: two equal differences make me treat it as linear, two equal second differences suggest quadratic, and three stable ratios hint at geometric growth; modular checks (parity, mod 3) are quick ways to kill bad ideas. I also prefer the model with the smallest integers in its description. With k-parameter fits I want at least k+2 terms; otherwise a cubic can always be made to hit five points, so uniqueness is illusory. If two simple rules survive, I say the data are insufficient and keep both hypotheses. Would you accept 100 here, or would you want another term before committing?