Mathematics with Minimum Raw Material, Part 2

The following are some simple mathematical problems discussed or
referrred to in Littlewood's miscellany (Edited by Bela
Bollobas, Cambridge University Press). In a section titled
Mathematics with minimum "Raw Material" , he talks about what
pieces of genuine mathematics fall under this category, before giving
some examples.
"A sine qua non is certainly that the result should be intelligible to
the amateur. We need not insist that the proof also should, though
most often it is."
(a) There are an infinite number of prime numbers.
A prime number is divisible only by itself or by unity. The proof due
to Euclid is as follows: Consider that we know k different
numbers to be prime. Let us denote them as follows, a,b,...,k.
It follows that if we multiply these prime numbers together, the
resulting number (ab...k) would not be a prime number. However,
(ab...k)+1 is a prime number since it is not divisible by any
of the prime numbers preceding it. Since the number of prime numbers,
k, is arbitrary and for every k known prime number there
is one more prime number, the number of prime numbers is infinite.
(b) An experiment to prove the rotation of the Earth.
Imagine a glass tube shaped in the form of a ring and filled with
water. If you place this tube flat on a table on the North Pole, then
it will be rotating along with the Earth at a rate of one revolution
every 24 hours. Now you take the tube and flip it suddenly by 180
degrees in the horizontal plane so that it is now upside down. Water is
now flowing (in a direction opposite to the rotation of the Earth)
round the tube at a rate of one revolution every 12 hours. This
movement of the water can be detected.
(c) Surfaces with just one side.
Take a ribbon and lay it flat on a table. Lift up each of its ends and
join them. You will have formed a ring, which has one outer surface
and one inner surface. Now lay the ribbon on the table and keeping one
end fixed, twist the other end, so that it is flipped (i.e the side
that was touching the table is now exposed and vice versa). Take the
two ends of this twisted ribbon and join them again to form a
ring. This ring has only one side.
You can convince yourself by running your finger along the surface of
the ring. You will find that you will return to where you started
from.
This ring is called a Mobius Strip.