Sequences — Number Pattern Reasoning

What this level trains

A number sequence puzzle gives you a run of numbers and asks you to work out the rule that generates them, then choose the next term. It trains inductive reasoning: abstracting a rule from limited examples and using that rule to predict what comes next.

The common rules fall into a few families: arithmetic and geometric sequences, difference sequences where the gaps themselves follow a rule, alternating sequences where odd and even positions run on separate tracks, and combination rules such as the Fibonacci type where each term is built from the previous ones.

The real difficulty is not calculation but knowing which family to suspect first. The planned eight levels raise the number of layers, from single-layer arithmetic and geometric early on to alternating and difference patterns, then second-order differences and named sequences like squares and cubes.

This level is still in development, so no difficulty parameters are published yet. It is a close cousin of the 24 game on this site: 24 trains building a target from given numbers, while number sequences train abstracting a rule from examples.

How to play

1. Look at the differencesSubtract each term from the next and see whether the gaps follow a pattern, such as constant, repeating or steadily rising.
2. Then look at the ratiosIf the differences look chaotic, divide each term by the previous one to check whether the sequence is geometric or nearly so.
3. Split the sequenceSeparate the odd and even positions into two tracks; many sequences that look baffling are really two simple sequences interleaved with each other.
4. Verify before answeringFeed your guessed rule back through every earlier term. Only if all of them fit should you choose an answer, which filters out most near-miss rules.

The planned 8 difficulty levels

The plan is eight levels: levels 1 and 2 use single-layer arithmetic and geometric rules; levels 3 and 4 introduce alternating patterns and first differences; levels 5 and 6 cover second-order differences, squares, cubes and combination rules such as the Fibonacci type; and levels 7 and 8 cross two rules at once, sometimes needing several rounds of differencing on paper before the structure shows. This level has not launched yet, so no time thresholds are listed.

1. Level 1A single-layer arithmetic sequence
2. Level 2A single-layer geometric sequence
3. Level 3Alternating rules on odd and even positions
4. Level 4First-order differences (the gaps themselves form a pattern)
5. Level 5Second-order differences and square numbers
6. Level 6Cubes and combined forms (such as Fibonacci-style)
7. Level 7Two rules interacting
8. Level 8Two interacting rules that only become visible after several rounds of differencing

Tips and common mistakes

• Start with the simplest guessTry differences, then ratios, and only then combination rules. Testing in that order noticeably raises your hit rate on number sequence puzzles.
• Group when there are many termsCheck the terms in groups of three or four; alternating rules usually reveal themselves at once when you look at them in groups.
• Write the differences downListing the gaps on paper or in a note is far more reliable than pure mental arithmetic and makes second-order structure much easier to see.
• Verifying beats guessingAlways check a rule against the earlier terms. Most mistakes are not failures to find a rule but rules that looked right and were never verified.

Frequently asked questions

Can a sequence puzzle have more than one answer?

With too little information, yes: you can always construct several rules that fit. A well-formed puzzle gives enough terms for the natural rule to be unique, usually five or six, and this site question bank will be filtered to that standard.

Do I need to memorise common sequences?

You do not need to memorise questions, but recognising the shapes of squares, cubes, Fibonacci numbers and triangular numbers helps a lot, because they are the most common building blocks for these puzzles.

How is this different from the 24 game?

The 24 game trains constructing a target from given numbers, a search-and-build skill. Number sequences train abstracting a rule from examples, which is induction. They sit at opposite ends of mathematical thinking.

What is a good way to start a hard sequence?

Begin with the cheapest checks: subtract each term from the next, then divide each by the previous one, and only after that look for interleaved tracks. Writing the gaps down turns most baffling sequences into an obvious structure within a couple of steps.

When will Sequences be available?

Sequences is still in development. Once it launches, this page will list the rule type and time thresholds for each of the eight levels. Until then, the method above, differences first and ratios second, works on almost any number sequence you meet.

Related levels

• 🃏 24 Game Level 4 · Mental Arithmetic / Number Sense / Working Memory
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• 🧠 N-Back Level 5 · Working Memory / Continuous Updating / Resistance to Interference