Games and Mathematics (22 page)

As if to illustrate that the variety of geometrical theorems is indeed endless, a curious small book was published in 1974 with this title.

The elegant in Figure
11.9
is almost self-explanatory. A chain of six circles touches the inside of another circle. (They could also touch instead on the outside.) Join the points of tangency of ‘opposite’ circles in the chain and the three lines are concurrent. As always happens with such theorems, we can imagine one or more of the circles ‘blowing up’ until it becomes a straight line. In Figure
11.10
, two of the circles have done so, and all six circles are external.

As the title says, the book was also about ‘other new theorems’ including the heptagon theorem, the three-conics theorem and the
nine
-circle theorem.

Where did these theorems come from? Did the three authors invent or discover them? Did they reason initially or start from experiment? We don't
know because they cunningly remark only that, ‘We derived a lot of enjoyment from evolving them’, adding that they have included rather a large number of diagrams ‘for their beauty of design, which we have found to have an appeal to many of our friends who are not mathematicians’ [Preface].

It is a commonplace that modern science creates vast numbers of beautiful images, and mathematics-as-science is no exception. All the miniature worlds of mathematics are exceptionally beautiful, and elementary geometry is only unusual in making that fact so very obvious.

We have seen that,
It is naturally tempting to wonder if you can add up the sequence of squares:
Yes, you can, and the standard result is
n
(
n
+ 1)(2
n
+1) which always annoys me because the factor 2
n
+ 1 seems out of place. Anyway, let's try to find the
sum of the odd squares instead, which is less well known. A
scientific
approach is to calculate the first few sums and try to spot a pattern:
Since the sum of the squares is the product of three factors, with an extra factor of 1/6, we take this to be a giant hint and write down the factors of each number in the right-hand column. The last three rows are especially suggestive. The sums to 11
²
and 13
²
include the factors 11 and 13 but the sum to 9
²
does not include the factor 9. Where can it have gone to? Put on your Sherlock Holmes thinking cap or puff on your favourite pipe, and the answer will appear. All we have to do is to pinch the factor 1/6 from the sum of all the squares:

The sum of
is
so the sum,
ought to be,
It is, though we haven't proved that conclusion. Before we leave this little experiment, we will make another observation: the factors 2 in 2
n
−1, 2
n
and 2
n
+ 1 suggest that we re-write the sum of the squares which is
n
(
n
+ 1)(2
n
+ 1), as
Aha! That annoying asymmetry has disappeared!

Many puzzles about the polygonal numbers are
relatively
easy to answer, plausibly because they show very strong patterns and, we might speculate, the stronger the pattern, the easier the proof. After all, proof depends on pattern, and it is by spotting patterns that ideas for proofs appear.

There is a lot to be said for that large assumption, which we will meet again later. Here we are going to test it, by sketching some problems raised by the prime numbers
.