# Number sequence solving strategies

Number sequences are one of the oldest puzzle categories in the book. You would find it hard to look through a days worth of puzzles posted here and to not find any puzzles that involve some sort of number sequence.

What are some strategies that you can use to solve a number sequence puzzle, and what are some common tricks/ideas used when making them to look out for?

After checking OEIS, the first thing to do is usually to try polynomial extrapolation by the method of differences:

1. Calculate the differences between successive numbers
2. See if all the results are the same.
• If they are, you have successfully fitted a polynomial to the sequence.
• If they are not, start again, using the differences as the number sequence.

The earlier you hit identical numbers, the more credible your polynomial fit is. If you have, say, three identical digits, you can be almost sure you got the intended solution. On the other hand, if you continue until you only have one number left, you have fitted an Nth order polynomial to N+1 data points, which is always possible, and the extrapolated value will have nothing to do with the intended solution.

If you find a credible match, just add one more of those identical numbers on the right, and work backwards to get the next number in the series.

Now then, if both OEIS and the polynomial fit fail to give an answer, then the situation is called a number sequence puzzle. Because many simple good ideas have already been had, check with the usual suspects first:

• self descriptive sequences
• getting the next number by manipulating the digits of the previous one in some way
• and so on, there are many good ones to be found in the tag.

If you still got nothing, then you either have an interesting, original puzzle, or a useless, unsolvable one. You'll need to figure out which one, in order to decide how many hours you are going to spend. There sure have been many gems on this site, but many of the other kind too.

I'm very bad at puzzles, but here's a simple and useful tip when making a number sequence.

A number sequence, no matter how original you think it is, is probably on OEIS. I'm not saying there aren't many great number sequence puzzles, but simple ones with one rule tend to be found by a search.

This puzzle is an example of a unique idea that unfortunately was not checked on OEIS before posting. Within 2 minutes, the OEIS link was posted in a comment, revealing the solution.

It may sound incredibly simple, but make sure your sequence is not on OEIS.

Now, on the other hand, you can reverse this and use it when you're solving a puzzle.

Many puzzles (but not ones on this site) are created to be a little brain teaser rather than an elaborate thought provoker. If you really want to solve it, type it into OEIS. It might just come up.

These might not really be tips, but it's all I know.

• this is great, and does answer my question, as looking up a sequence in OEIS is a valid way of attempting to solve number sequence puzzles Oct 29, 2018 at 23:20