The New York Times recently posted a piece on problem solving which asked readers to first solve a problem:
"We’ve chosen a rule that some sequences of three numbers obey — and some do not. Your job is to guess what the rule is.
We’ll start by telling you that the sequence 2, 4, 8 obeys the rule."
You can test your hypotheses by typing sequences into three boxes to see if they follow the unstated rule. Once you think you know, you type in a description. Most people it turns out, suggest an answer without ever trying a sequence that returns a firm "NO." Psychologists interpret this as being evidence of confirmation bias: once we get a "yes" for our theory - we don't poke around trying to find a "no."
When I teach chemical kinetics, I point out to students that few experiments can prove a reaction goes in a particular sequence, only that the data is consistent with a proposed mechanism. No answers can be as or more critical to problem solving as yes.
I failed to 'correctly' solve the puzzle, [SPOILER ALERT] though I did get several no answers. One rule I tried was an: 21, 22, 23 = 2, 4, 8. The sequence 1, 1, 1 follows that rule (11, 12, 13 are all one), but yielded a no. The rule an = 2 x an-1: 2, 2x2=4, 2x4 worked for every sequence I tried, but is not 'the 'answer. The answer is that correct sequences have each number larger than the last.
The study suggests I failed not only because of confirmation bias, but because I complicated the problem, assuming that there was some sort of trick to the rule. Actually, I assumed the technical mathematical meaning of sequence held, in that there was a rule that uniquely specified each number in the sequence given the starting value(s). An ordered list of numbers, each of which is larger than the previous value is not a sequence in the mathematical sense.
In retrospect, I should have tried the sequence 0, 0, 0. It follows the rule I proposed (an = 2 x an-1) as the correct one, but returns a "no." It would have ruled out my proposed rule, a useful "no". (I might also have tried non-integer numbers.) I failed in part because I didn't understand the question they were asking, we didn't have the same definition of "sequence." In some sense I fell prey to the "when all you have is a hammer, everything looks like a nail" scheme.
There are more than 2500 rules that would give you the mathematical sequence 2, 4, 8. See Sloane's encyclopedia of integer sequences. My first proposed sequence is A000079 in the collection.
For more about sequences and Sloane's encyclopedia, read this article at AT&T.
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