MathDB

a) Several black squares of side

1

cm are nailed to a white plane with a nail of thickness

0 . 1

cm so that they form a black polygon. Can it happen that the perimeter of this polygon is

1

km long? (The nail is not allowed to touch the boundary of any of the squares . ) (b) The same problem as in (a) but with a nail of thickness

0

(a point ) . (c) Several black squares of side

1

cm lie on a white plane so that they form a black polygon (possibly having more than one piece and/ or having holes) . Can it happen that the ratio of its perimeter (in centimetres) to its area (in square centimetres) is more than

100000

?
(Hungarian Folklore)

Problem Statement