On I.Q. tests and in other places, one is often confronted with problems of the form: “What’s the next element in the following series: 1, 4, 9, …”
Technically, such questions have no right answer, because there are a multitude of ways to generate the initial elements of the series, and each way can produce a different result for how the series should continue. What is being sought is the generator for the series that is somehow the simplest, cleverest, most obvious, or most elegant. There is an esthetic at work in determining the preferred solution.
For example, for the series above, an obvious answer is 16, because the initial elements of the series are the squares of the first three natural numbers, and so the obvious way to continue the series is with the squares of the subsequent natural numbers. The series is given by sn = n2.
A similar type of question involves a mapping between a series of expressions and values. In that case, the preferred answer...
There's a famous (and probably apocryphal) story about the mathematician Carl Friedrich Gauss that goes something like this:
Gauss was 9 years old, and sitting in his math class. He was a genius even at this young age, and as such was incredibly bored in his class and would always goof off and get into trouble. One day his teacher wanted to punish him for goofing off, and told him that if he was so smart, why didn't he go sit in the corner and add up all the integers from 1 to 100? Gauss went and sat in the corner, but didn't pick up his pencil. The teacher confronted him, saying “Carl! Why aren't you working? I suppose you've figured it out already, have you?” Gauss responded with “Yes – it's 5,050.” The teacher didn't believe him and spent the next ten minutes or so adding everything up by hand, only to find that Gauss was right!
So how did Gauss find the answer so fast? What did he see that his teacher didn't? The answer is simple, really – it's all about pattern...