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Function machine

Numbers go in, numbers come out. From a few pairs, find the rule — ×3 + 2, square it, reverse the digits — and say what comes out next.

  • Pattern
  • Timed
  • Endless levels

How it plays

Each line is a number going into the machine and the number that comes out. Find what the machine does and give the output for the new input — or, if you turn it on, run it backwards and give the input.

Kinds of machine unlock as you climb: add, multiply, multiply then add, take away from a number, square then add, multiply by the next number up, cube, add then square, reverse the digits, add the digits, multiply the digits. Then the numbers grow and fewer pairs are shown.

Every kind of machine is tried on the pairs; the question is only asked when every kind that fits gives the same answer. The wrong options are what a rule fitted to only one or two pairs would give.

Keys

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

What it trains

This practises inferring a rule from examples and checking it against all of them. It is practice for function and rule questions, nothing wider.

Learn

How to get better at function machine: 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. Read every pair before you guess. The inputs are listed from smallest to largest, so you can see how fast the outputs grow.
  2. Compare neighbouring pairs: how much does the output change for each step of the input? Steady, growing or falling tells you the family of machine.
  3. Name one rule and run it on every pair shown, not just the first two. One pair that fails kills it.
  4. For “What comes out”, run the rule on the new input. For “What went in”, undo the rule’s steps in reverse order: undo the add first, then the multiply.

Techniques and tricks

  • Change over change gives the multiplier

    For any “multiply, then add” machine, divide the difference in outputs by the difference in inputs. That is the multiplier; output minus multiplier × input is the number added.

    e.g. 10 → 64, 13 → 79: 15 ÷ 3 = 5, and 64 − 50 = 14. The machine is × 5 + 14.

  • Run it backwards, last step first

    Undo each step with its opposite, starting from the step done last.

    e.g. × 5 + 14 gave 74: 74 − 14 = 60, then 60 ÷ 5 = 12. So 12 went in.

  • Falling outputs: check the sum

    If outputs fall as inputs rise, add each input to its output. A total that never changes means “take it away from” that total.

    e.g. 20 → 50 and 43 → 27 both make 70. So the machine is 70 − x, and 69 → 1.

  • Compare with the square

    When the output gaps grow, set each output beside the square of its input. A fixed distance from the square is “square it, then add”.

    e.g. 11 → 108, 13 → 156: 121 − 13 and 169 − 13. So 4 → 16 − 13 = 3.

  • Every output a square: take roots

    If every output is a perfect square, take its square root and compare that with the input. A fixed gap means “add, then square”.

    e.g. 7 → 121, 10 → 196: the roots 11 and 14 are each 4 more. So 169 = 13² came from 9.

  • Digit machines

    When outputs bear no steady link to the size of the input, look at its digits: reversed (perhaps plus a number), added up (perhaps times a number), or multiplied together.

    e.g. 52 reversed is 25, + 3 = 28. 47: (4 + 7) × 3 = 33. 47: 4 × 7 = 28.

  • Know the landmark numbers

    Squares to 20², cubes to 10³ (1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000) and n × (n + 1) (6, 12, 20, 30, 42, 56, 72, 90). Seen in the outputs, they name the machine at once.

    e.g. 343 is 7³; 56 is 7 × 8; 196 is 14².

As the levels climb

  • You start with add and multiply, four pairs shown. Next come “multiply, then add” and “take it away from”, then bigger numbers, then three pairs, then 80% of the time.
  • Square-then-add and “multiply by the next number up” arrive next, then bigger numbers again, then only two pairs (more appear only when two cannot fix the rule), then 65% of the time.
  • With two pairs there is no spare to check against, so the change-over-change trick and the landmark numbers do the work.
  • Then cube, add-then-square, reverse the digits and add the digits arrive together; after bigger numbers again, multiply the digits. After that only the numbers grow, without end.
  • “What went in” questions get five seconds more and never use add-the-digits or multiply-the-digits, since many inputs give the same output there.

Traps to avoid

  • A rule from one pair. 7 → 28 alone could be + 21 or × 4. The options include what such a rule would give, so check your rule against every pair.
  • A rule that fits two pairs and breaks on the third. 1 → 6 and 2 → 9 fit × 3 + 3, but 3 → 14 does not: that machine is square, then add 5, and 4 → 21.
  • One off. The answer ± 1 is often among the options, to catch a slip in the last step. Redo the final sum before you choose.
  • Undoing in the wrong order. For × 5 + 14 and an output of 74, dividing first gives 14.8 — not a whole number, and wrong. Take away the 14 first.

How to practise

  • Turn on “What went in” under “Questions” once going forward is easy. Undoing a rule shows whether you really know it.
  • Switch “Answer by” to “Typing the number”. With no options to test, you must find the rule itself.
  • Raise “Options shown” towards 8: each wrong option is a rule that fits only some pairs, and finding the pair it fails is the skill.
  • After a miss, read the “Check” line in the working and run the machine on one pair yourself before the next question.

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.

More pattern

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