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'Pick's Quadratics' printed from http://nrich.maths.org/
To prove Pick's Theorem does not require any advanced mathematics,
just careful reasoning. The
Proof of Pick's Theorem
is the challenge in the next problem
which leads you step by step through the proof. This problem
is about a generalistaion of Pick's Theorem.
Pick's Theorem can be generalised as follows:
'For any planar polygon with vertices at lattice points the
quadratic formula $i(k)=Ak^2 - Bk +C$ gives the number of
$k$-points inside the polygon and the quadratic formula $g(k)= Ak^2
+ Bk +C$ gives the number of $k$-points in the closed polygon
(including the boundary and the interior points), where $A$ is the
area of the polygon.'
This challenge asks you to verify this generalised form of Pick's
Theorem for a particular rectangle.
The proof that the given quadratic formulae hold for all polygons
is difficult and requires mathematics beyond school level. However
it is worth noting that this is the form of Pick's Theorem that
generalises to 3 and higher dimensions.