What comes next — and why that is a fair question
Number sequences where the continuation is not a matter of taste. Every sequence is offered to all seven rules this puzzle knows, and thrown away if two of them fit the numbers shown while disagreeing about the next one.
Generated on this device
Why most "what comes next" puzzles are unfair, and what is done about it here
Because no finite list of numbers has only one continuation. Given 2, 4, 6, 8 there is a polynomial through those four points ending in absolutely any fifth value you care to name. A puzzle that insists the answer is 10 and marks 11 wrong is asserting a preference and calling it a fact.
So no unique continuation is claimed here. What is claimed is narrower and actually checkable: of the seven rules this puzzle knows, exactly one fits the numbers shown. Every candidate sequence is offered to all of them — constant step, constant multiplier, two steps taken in turn, a step that grows, two runs woven together, add-a-number's-own-digits, and add-the-previous-two — and if two fit the visible terms while disagreeing about what follows, that sequence is thrown away rather than published.
The example that makes the case is 1, 2, 4, 8, 16. Doubling says 32. But 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16 — adding a number's own digits — says 16 + 1 + 6 = 23, and fits every term just as well. That sequence is genuinely ambiguous, so it never appears here. The test suite checks that specific case, because it is the clearest illustration of the failure the check exists to prevent.
Naming a rule this tool has never heard of will still beat it. That is a real limit, and it is stated rather than hidden.
The wrong answers are not random
Random numbers alongside the right one make a multiple choice trivial — the plausible-looking option is the answer. Every wrong option here comes from a particular way of misreading the rule: keeping the last step where the step itself changes, continuing the row of differences instead of the numbers, taking the step from the wrong place, adding where you should multiply.
That is why choosing one tells you something, and why the explanation names what happened rather than saying only that it was wrong.
How to work one out
Write the gaps under the numbers first. If they are all the same, you are done. If they are not, write the gaps between those — a constant second row means the step is growing by a fixed amount, which covers the square numbers, the triangular numbers and a good deal else.
If neither row settles, check whether the gaps alternate between two values, and then whether the sequence is really two sequences taken in turn. The tell for the second is that every other number is doing something sensible on its own while the whole thing looks like nonsense.
When the numbers grow much faster than the gaps can explain, look for multiplication, or for each number being built from the ones before it.
What this is not
Not a test of anything. Sequence puzzles have a long association with intelligence testing, and that framing is both culturally loaded and unsupported: how quickly you spot that the gaps are growing by three says something about how much practice you have had with sequence puzzles, and nothing else. There is no score kept, nothing is stored beyond the current page, and no claim is made about what any of it measures.
Related
Logic Grids and Codebreaker for deduction with a definite answer, or the puzzle hub for the rest.
Questions people actually ask
Isn't "what comes next" always a bit of a con?
Usually, yes — no finite list of numbers has only one continuation, because a polynomial can be drawn through any terms and end anywhere. So no unique continuation is claimed here. What is claimed is that of the seven rules this puzzle knows, exactly one fits the numbers shown; every candidate is offered to all seven and discarded if two fit the visible terms while disagreeing about the next one.
Give me an example of one you rejected.
1, 2, 4, 8, 16. Doubling says 32. Adding a number's own digits to itself — 1+1=2, 2+2=4, 4+4=8, 8+8=16 — says 16+1+6=23, and fits every term just as well. Genuinely ambiguous, so it never appears. The test suite checks that exact case.
What if I have a rule you have not thought of?
Then you will beat the check, and you would be right to. The guarantee is bounded by the seven rules in the library and that bound is stated rather than hidden — claiming more would mean telling people who reasoned correctly that they were wrong.
Why are the gaps shown under the numbers?
Because subtracting consecutive terms is arithmetic, not the puzzle. Hiding them makes the thing slower without making it more interesting, and anyone who solves these regularly writes them down first anyway.
Are the wrong answers random?
No. Each comes from a specific way of misreading the rule — keeping the last step where the step changes, continuing the row of differences instead of the numbers, taking the step from the wrong place, adding where you should multiply. That is why picking one tells you something, and why the explanation names what happened.
Does this measure anything about me?
No, and nothing here suggests it does. Sequence puzzles have a long association with intelligence testing that is both culturally loaded and unsupported. How fast you spot that the gaps grow by three reflects how much practice you have had with sequence puzzles. No score is kept, and nothing is stored beyond the page you are on.