This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
Are "complete this series" problems well defined?
Since there are infinitely many formulas that will fit any finite
series, many people object that such problems have no good answer.
But isn't this a special case of the general observation that theory
is underdetermined by experience (in other words, that there are a
lot of world views that are consistent with all the facts that we
know)? And if so, doesn't this objection really apply to all puzzles?
Isn't it just more obvious in the case of series puzzles?
As a long-time observer of rec.puzzles nit-picking, I have never seen
a puzzle answer that could not be challenged. The list of assumptions
made in solving any puzzle is neverending. Luckily, most of us share
all or nearly all of these assumptions, so that we can agree on an
answer when we see it.
All of this has a lot to do with topics such as computational
complexity, algorithmic compressibility, Church's thesis, intelligence