Game Guide
Walk the five Sequence demo levels with one diagnostic order: first differences, then ratios, then second differences, then sums of prior terms.
Sequence asks one question five times: what rule produced these numbers? The answer is never guessable in the useful sense, but it is almost always findable, because the space of rules that puzzle designers actually use is small. There is a fixed order in which to test for them, and running the tests in that order turns the game from inspiration into procedure.
The five free browser levels are a good sample because they draw on four different rule families, including one that catches nearly everybody. Here is the diagnostic order first, then the levels.
Here is the MyRin short for Sequence. Watching somebody run the difference test across a row is the fastest way to see what the method looks like in practice.
Run them in that order and stop at the first one that works. The order is not arbitrary: it goes from cheapest and most common to most expensive and least common.
Choices 5, 6, 7, 8. First differences on the visible terms: 4 minus 2 is 2, and 10 minus 8 is 2. Constant, so the rule is plus two each step. The missing term sits between 4 and 8, so it is 6.
Note what made this trivially checkable: the terms on both sides of the gap agree. If the answer were 7, the sequence would step by 3 and then by 1, and nothing would be constant. The gap in the middle is a constraint from both directions, and using both sides is the habit to build.
Choices 10, 11, 12, 13. First differences: 3, 3, then a gap, then 15 minus the missing term. Plus three each step gives 12, and 15 minus 12 is 3, which closes the pattern.
These first two levels exist to make the difference test automatic. Both are arithmetic, both resolve on the first test, and neither needs anything else. Test one alone will carry you through a large share of the easy levels in the installed game.
Choices 14, 15, 16, 17. First differences are 3 and 5, not constant, so test one fails and you move on. Ratios are 4 and 2.25, not constant, so test two fails. Second differences: 5 minus 3 is 2, which is the quadratic signature.
In practice you will probably shortcut this one by recognition: 1, 4, 9, 25 are the squares of 1, 2, 3 and 5, so the missing term is 4 squared, which is 16. Recognition is fine and faster, but run the second difference test at least once here anyway, because the installed game uses quadratic sequences that are not recognisable squares, and then the test is all you have. The same difference tests carry most of the Sangaku tablets too.
Choices 20, 25, 30, 35. Differences are large and shrinking, which points at a ratio rather than a difference. Dividing: 100 to 50 is halving, and 12 to 6 is halving. So the rule is approximately divide by two, and the missing term between 50 and 12 is 25.
The interesting part is that 25 halved is 12.5, not 12. The sequence is deliberately rounded, and the hint in the game says as much: divide by two, roughly. That is worth sitting with, because it teaches something real about this genre. A rule can be right while the terms are imperfect, and demanding exactness would lead you to reject the correct rule. Check that a rule explains the shape of every term rather than matching every term to the decimal.
Choices 4, 5, 6, 7. First differences are 0, 1, 1, and later 5, so not constant. Ratios are 1, 2, 1.5, not constant. Second differences are not constant either. All three earlier tests fail, which is itself the diagnostic: you are in the Fibonacci family.
Check the rule directly. Is each term the sum of the two before it? 1 plus 1 is 2. 1 plus 2 is 3. So the missing term is 2 plus 3, which is 5, and the check on the far side holds: 3 plus 5 is 8, and 5 plus 8 is 13.
The double 1 at the start is the tell. A sequence that opens with two identical terms and then grows faster than arithmetic but slower than geometric is almost always additive in this way. Once you have seen it, you will recognise it instantly, which is most of the value of meeting it in a demo.
Staring. Given a row of numbers, the instinct is to look at them and wait for the pattern to announce itself, and when it does not, to keep looking. Sequence rewards the opposite behaviour: compute something immediately. Even if first differences turn out to be the wrong test, having them written down changes what you can see, and the failure narrows the search rather than wasting it.
The second most costly habit is answering from one side of the gap. It is easy to notice that 3, 6, 9 steps by three, fill in 12 and move on without ever checking against the 15. On demo levels that is harmless. In the installed game, sequences that interleave two rules at alternating positions will punish it every time, and the only defence is verifying against the terms after the gap as well as before it.
Pattern recognition of this kind is one of the better studied components of fluid reasoning, and this game is a fairly pure instance of it: little arithmetic load, no memory load, no time pressure in the demo, just rule inference. If the five levels go quickly, the installed version carries 600 of them with the difficulty coming from less obvious rule families rather than bigger numbers.