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Number sequences

Find the rule behind a run of numbers and carry it one step on — adding, doubling, interleaving, growing gaps — with rules and numbers that keep coming.

  • Pattern
  • Timed
  • Endless levels

How it plays

A few numbers follow a rule; work out the rule and give the number that comes next — or, at higher levels, the one missing from the middle.

New kinds of rule unlock as you climb: a constant step, a constant ratio, two sequences taking turns, gaps that grow, each term the sum of the two before, alternating operations, squares and cubes. Then the numbers get bigger and fewer terms are shown.

After every answer the rule is stated in words. If you found a different rule that also fits, you have found something real — but the one stated is the one the sequence was made with.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises spotting regularities in numbers. Pattern puzzles like these are practice for themselves; they are not a route to a higher intelligence score.

Learn

How to get better at number sequences: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Look at the shape first: rising slowly, rising fast, falling, or zig-zagging. Fast growth points to multiplying; a zig-zag points to two sequences taking turns.
  2. Write the gaps between neighbours. Equal gaps mean add or subtract a constant.
  3. If the gaps are unequal, try each term divided by the one before. Equal ratios mean multiply.
  4. If neither works, try the gaps of the gaps, every other term, the sum of the two before, and the squares or cubes near each term.
  5. Check the rule on every term shown, then carry it on. With the ? in the middle, check your answer against the term after it too.

Techniques and tricks

  • Gaps of gaps

    If the gaps grow, write the gaps between the gaps. A steady second gap means the next gap grows by that much again.

    e.g. 3, 5, 9, 15, 23, ? — gaps 2, 4, 6, 8, so the next gap is 10: 33.

  • Every other term

    When the terms jump up and down, read the 1st, 3rd, 5th as one sequence and the 2nd, 4th, 6th as another. Each has its own step.

    e.g. 4, 30, 6, 28, 8, 26, ? — one adds 2 (4, 6, 8), the other takes 2 (30, 28, 26). Next: 10.

  • Sum of the two before

    If each gap equals the term two places back, each term is the sum of the two before it.

    e.g. 3, 5, 8, 13, 21, ? — 13 + 21 = 34.

  • Look for a nearby square

    Set each term beside the square (or cube) of 1, 2, 3, 4 … A fixed distance from those squares names the rule.

    e.g. 6, 11, 18, 27, 38, ? — that is 2² + 2 up to 6² + 2, so the next is 7² + 2 = 51.

  • Ratio nearly steady: multiply, then add

    When each term is about double (or triple) the one before but a little off, find the leftover: the rule is multiply, then add or subtract a small number.

    e.g. 4, 9, 19, 39, ? — each is double the last plus 1. 39 × 2 + 1 = 79.

  • Two operations taking turns

    When the gaps follow no pattern, check whether the steps alternate between adding and multiplying.

    e.g. 2, 5, 10, 13, 26, ? — + 3, × 2, + 3, × 2, so next is + 3: 29.

  • Halfway for a middle gap

    In an add or subtract sequence, a missing middle term is the average of its two neighbours.

    e.g. 22, 35, 48, ?, 74 — (48 + 74) ÷ 2 = 61.

As the levels climb

  • Rules arrive one at a time: add, subtract, multiply, two sequences interleaved, growing gaps, the sum of the two before, add-then-multiply, squares and cubes, multiply-then-add. At least half the questions use the newest rule, so expect it.
  • Early on the terms shown drop from six to five, then — with “Where the ? goes” on “At the end, then anywhere” — the ? moves into the middle; later only four terms are shown.
  • Between new rules the numbers grow without end: bigger starts, bigger steps, more digits. Hold the rule, not the numbers.
  • The clock never tightens here. Each new rule and each size step adds time; what gets harder is the rule and the fewer clues.

Traps to avoid

  • The repeated gap. In a growing sequence the last gap repeated is offered: for 3, 5, 9, 15, 23 that is 23 + 8 = 31. The gap grows again, to 33.
  • The repeated ratio. In an add sequence, the last ratio repeated is offered: for 22, 35, 48 that is 48 × 48 ÷ 35, about 66. Check gaps before ratios.
  • The sum of the last two. It is often offered — 35 + 48 = 83 for 22, 35, 48 — and looks right if you half-remember the sum-of-two rule.
  • One off. The answer ± 1 is often among the options. Redo the last step before choosing.

How to practise

  • Set “Where the ? goes” to “Always at the end” to learn new rules, or to “Anywhere” to practise middle gaps from the start.
  • Switch “Answer by” to “Typing the number”. With no options you cannot check candidates against a list; you must find the rule.
  • Lower “Options shown” to 2 while a rule is new; raise it towards 8 to face every lure at once.
  • After every answer, read the stated rule and say it in your own words. If you found another rule that fits, check it on the next question of that kind.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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