Solving number sequence puzzles
Every sequence question has a rule. The hard part is not arithmetic — it is choosing which rule to test first. Here is an order that works for most puzzles.
Step 1 — write the gaps
Subtract each term from the next one. If the gaps are constant, the rule is addition:
3, 7, 11, 15, ? → gaps are 4, 4, 4 → next is 19.
Step 2 — check the ratios
Divide each term by the previous one. If the ratios are constant, the rule is multiplication:
2, 6, 18, 54, ? → ratios are 3, 3, 3 → next is 162.
Step 3 — check the gaps of the gaps
If the differences are not constant but they change steadily, the rule is a growing gap:
2, 5, 9, 14, ? → gaps 3, 4, 5 → next gap 6 → 20.
Step 4 — look for sums
Fibonacci-style rules add the previous two terms:
3, 5, 8, 13, ? → 8 + 13 = 21.
A quick test: is each term roughly 1.6× the previous one? That ratio is the golden ratio, the signature of Fibonacci-like growth.
Step 5 — check squares and cubes
When terms grow fast but irregularly, compare against squares:
5, 10, 17, 26, ? → these are 2²+1, 3²+1, 4²+1, 5²+1 → next is 37.
Step 6 — check alternating rules
If no single rule fits, split the sequence into odd and even positions and test each half separately. Sequences like 9, 4, 11, 6, 13, 8 alternate “+2” for odd positions and “+2” for even positions.
Quick checklist
- Gaps constant? → addition
- Ratios constant? → multiplication
- Gaps growing steadily? → growing difference
- Each term ≈ sum of previous two? → Fibonacci
- Terms near perfect squares? → squares plus offset
- Nothing fits? → split odd/even positions
Common mistakes
- Assuming the rule must be arithmetic when it is multiplicative.
- Ignoring negative or zero terms in a plausible rule.
- Stopping at the first rule that fits three terms — always verify against all given terms.