Exercise 3: Why the Real Pólya Case Is Stronger Than an Abstract Statement — Possible Solution
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An abstract statement like "empirical checking isn't proof" is easy to
nod along with in the abstract, but it doesn't convey how far
empirical evidence can genuinely go before it fails - it's easy to
imagine that a pattern breaking down would happen quickly, after only
a handful of exceptions, if the pattern were ever going to fail at all.

The real Pólya conjecture case makes the danger concrete and specific:
the conjecture held for every number mathematicians checked for
decades, and even after Haselgrove proved in 1958 that a counterexample
had to exist somewhere, no one could actually produce one smaller than
his own enormous estimate (around 1.845 x 10^361) for two more years.
The real smallest counterexample, n = 906,150,257, wasn't found until
1980 - meaning the conjecture looked true for an genuinely enormous
range of numbers before finally, concretely failing.

This shows that "it held for a very large number of checked cases"
provides essentially no real guarantee about what happens at the next
case, or the case after that, or the case a hundred million further
along - a pattern can survive checking against numbers in the hundreds
of millions and still be false. An abstract warning doesn't convey that
scale; a real, dated, numbered historical example does.

ANSWER: The real Pólya case is stronger because it shows concretely,
with real numbers and real dates, just how far a false pattern can
survive empirical checking (holding for an enormous range of numbers
before failing at n = 906,150,257) - which makes the abstract warning
"empirical checking isn't proof" tangible and memorable in a way a
general statement alone can't.

WHY THIS WORKS AS AN ANSWER
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This explains the real pedagogical/persuasive difference between an
abstract principle and a concrete historical case, using the chapter's
own specific numbers (the scale of checking, the real counterexample)
as the evidence for why concreteness matters here.