MathDB

Let

M

be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point

P=(x,y)\in M

we call the points

(x-1,y)

(x+1,y)

(x,y-1)

(x,y+1)

neighbors of

P

. Let

S

be a finite subset of

M

. A one-to-one mapping

f

S

onto

S

is called perfect if

f(P)

is a neighbor of

P

, for any

P\in S

. Prove that if such a mapping exists, then there exists also a perfect mapping

g:S\to S

with the additional property

g(g(P))=P

for

P\in S

Problem Statement