MathDB

A set

A

of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to

A

there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number

k

and such a way of moving the kings that after

k

moves the kings will occupy all squares of the chessboard? Consider the following cases:
(a)

A

is the set of all squares, both of whose coordinates are multiples of

100

. (There is a horizontal line numbered by the integers from

-\infty

+\infty

, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.)
(b)

A

is the set of all squares which are covered by

100

fixed arbitrary queens (i.e. each square covered by at least one queen).
Remark: If

A

consists of just one square, then

k = 1

and the required way is the following: all kings to the left of the square of

A

make one move to the right.

Problem Statement