MathDB

Let

P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}

be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length

1

\sqrt 2

, so that both ends of the segment are in

P

and do not intersect each other. Furthermore, for each point

(a, b)

it is true that i) if

a + b

is a multiple of

3

, then it is an endpoint of exactly

3

segments. ii) if

a + b

is an even not multiple of

3

, then it is an endpoint of exactly

2

segments. iii) if

a + b

is an odd not multiple of

3

, then it is endpoint of exactly

1

segment. a) Check that with

n = 6

it is possible to satisfy all the conditions. b) Show that with

n = 2015

it is not possible to satisfy all the conditions.

Problem Statement