MathDB

Given a chess-board

99\times 99

with a set

F

of fields marked on it (the set is different in three tasks). There is a beetle sitting on every field of the set

F

. Suddenly all the beetles have raised into the air and flied to another fields of the same set. The beetles from the neighbouring fields have landed either on the same field or on the neighbouring ones (may be far from their starting point). (We consider the fields to be neighbouring if they have at least one common vertex.) Consider a statement:
"There is a beetle, that either stayed on the same field or moved to the neighbouring one".
Is it always valid if the figure

F

is:
a) A central cross, i.e. the union of the

50

-th row and the

50

-th column?
b) A window frame, i.e. the union of the

1

-st,

50

-th and

99

-th rows and the

1

-st,

50

-th and

99

-th columns?
c) All the chess-board?

Problem Statement