Thinking outside the box or just psychic?

[Don]

Number pattern, such as 1, 2, 3, 4, 5, 6, ? seems to have certain importance to humans since we see them in math and IQ tests. Today I was looking at some more complicated ones, and it made me wonder does seeing a pattern mean anything about how you're thinking or whether you just happen to be thinking the same as whoever gave the question. For example

10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, ___

I'll go out on a limb and say that if you haven't seen this one before, you'll not get the answer, which is 31. This is the sequence of numbers you get when you take 16 from base 16 and then successively lower bases. Now if you didn't get it without it explained, like I did, someone is supposed to say that you didn't think outside the box, that you're limited to base 10. So I decided to make it a bit more complicated. Instead of taking just 16, we'll do the sequence 16, 17, 18... with the same criteria, and then you get

10 12 14 16 18 __ 22 25 30 34 42 52 130 1002 11110

We'll discard the terms after 52 since the large numbers could possibly give away the fact that we're messing around with bases here.

10 12 14 16 18 __ 22 25 30 34 42 52

Now I'm wiling to bet that anyone doesn't know you started with base 16 and subtract 1 base each term will never be able to guess the missing term. Why did I leave out a term? Because it's 1A (21 in base 11) which would give away the fact that we're changing bases here.

Actually, why don't we get rid of all the numbers after 25, then we have:

10 12 14 16 18 __ 22

Now here's a pattern you can't possibly answer correctly! If you say it's 20 I say it's 1A, and if you say anything but 20, you're obviously dumb and didn't see that we're just adding 2 at a time!

I do like pattern puzzles, but it seems to me some of the 'harder' ones aren't really about how you can think. It's more about how you happen to think whatever the author was thinking at that point. I could make a pattern based on the number of posts everyone has on this board. No one outside of this board would have a remote chance of following it (and probably not even anyone on this board without being told), but it is, IMO, no less valid than somehow knowing that we start with base 16 and the base goes down by 1 each term.

[Kupek]

If someone answers with an equation that fits the given numbers, then that answer is correct, even if it's not what the authors had in mind. Further, if someone eliminates a significant portion of the tail of the sequence, then it's a pointless exercise. What your example did was basically try to hide the real behavior of the sequence, which eliminates the point of the exercise.

However, the process itself is similar to coming up with a scientific theory to explain physical phenomenon. If we can recognize that data from an experiment fits an equation, than we can actually learn something about the physcial world.

[Nev]

Don, I think you need to get out more.

[Don]

From what I understand intelligence tests aren't interested in your explanation of the numbers. Personally I think using stuff like changing bases is barely one step above as defining your own function anyway.

Most of such sequences are constructed delibrately to hide the trick. For example in the first one I made which you can find online by searching the sequence, the fact that the last term is 10000 (16 in base 2) and 3rd to last term 100 (16 in base 4) is usually omitted because these numbres are too nice and round and could give away the fact that bases are changing. Even without cutting out an unsensible amount of numbers, the last sequence I gave would be nearly impossible without the 1A term while leaving it in and substituteing any other term out would have made the problem at least doable because A is obviously not a base 10 term. It is also rather convenient that the original sequence doesn't have any term that requires using a number greater than 9 to create the illusion of base 10.