How to spot the pattern in a number sequence

Find the rule, predict the next term. The trick is knowing which questions to ask, and in what order.

Sequence puzzles reward a checklist more than inspiration. Run through the possibilities in order of how common they are.

The checklist

Differences. Subtract each term from the next. If the differences are constant, the rule is arithmetic. If the differences themselves form a simple sequence, take differences again — two rounds catches most quadratic patterns.

Ratios. Divide each term by the previous. A constant ratio means a geometric sequence.

Sums of previous terms. Fibonacci-like rules, where each term is the sum of the two before it.

Position-based. Compare each term against its index: squares, cubes, triangular numbers, or a simple formula in n.

Interleaving. If nothing fits, try reading alternate terms. Two sequences woven together look chaotic as one.

The ambiguity problem, and how it is handled

Strictly, any finite sequence can be continued by infinitely many rules — a polynomial can be fitted through any set of points. Puzzles that ignore this produce arguments, because a solver with a different but equally valid rule is told they are wrong.

Here the generator tests each candidate sequence against a library of rules and discards it if more than one fits. So 1, 2, 4, 8, 16 is never used: it continues as 32 by doubling and as 31 by the circle-chord rule, and both are defensible. A sequence you are shown has one rule from the library that works.

Working backwards

When the front of a sequence is confusing, start at the end. Later terms in a growing sequence are often more distinctive — a large number that is one less than a power of two is easier to recognise than the small terms that preceded it.

Frequently asked questions

Could more than one rule fit?

In principle yes for any finite sequence, which is why the generator rejects sequences where more than one rule from its library fits. Sequences like 1, 2, 4, 8, 16 are never offered because they are genuinely ambiguous.

What should I try first?

Differences between consecutive terms. If those are not constant, take differences of the differences. Between them these catch the majority of sequences.

What if the numbers jump around?

Try reading alternate terms — two interleaved sequences look random when read straight through.