MathDB

Let

\Lambda

denote the set of points

(x,y)

in 2D space with integer coordinates such that

0\leq x\leq 4

and

0\leq y\leq 2

. That is,

\Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}.

Find the number of ways to connect points of

\Lambda

with segments of length

\sqrt{2}

\sqrt{5}

such that the interior of any unit square with vertices in

\Lambda

contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct). [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2)); draw((0,0)--(1,1)); draw((0,2)--(2,1)); draw((1,1)--(2,0)); draw((2,0)--(3,2)); draw((3,1)--(4,2)); draw((3,0)--(4,1)); [/asy]

Problem Statement