I just found a page on "How to Find a Formula for a Set of Numbers". It's a cool little procedure for taking a series, like:
2, 8, 9, 11, 20
and producing a polynomial to give you the next ones in the series, like:
n^3^- 17/2 n^2^+ 49/2 n - 15
where n is the term number, starting from n=1. Try it out! Anyway, it was a method I learned in high school math league, and thought it was so cool I wrote a BASIC program on the old TRS-80 computers to do it. I had forgotten how to do it, and it was fun to see it again. I particularly liked the comment on the page:
"""If someone gives you the sequence, say, "1, 4, 9, 16", you could run them through the above process and get the answer that the person is probably looking for: the rule is n^2^ so the next value is 25. But you could also invent any number as the next number in the sequence, say 42, and come up with a rule for "1, 4, 9, 16, 42". Feel free to work it out. It comes out to:
17/24 n^4^ - 85/12 n^3^ + 619/24 n^2^ - 425/12 n + 17
and the next term is then 121.
So if you want to be obnoxious, the next time you are given a quiz of "find the next number in the series" problems, just pick any number you like and fill it in, and you'll be completely correct. You'll probably get a failing grade on the test, but you can enjoy the smug satisfaction of knowing you were right."""
I knew a kid who, because of a ridiculous fluke, had to redo some of his middle-school competency tests in high school. So, when presented with a series like 2,4,6,8,... he did this on a test (and yes he did fail the test and have to redo it). He was also shown a number of clocks, and asked what time does this show, and for all of the answers put "analog time".