MathDB

Let

n

be a fixed positive odd integer. Take

m+2

distinct points

P_0,P_1,\ldots ,P_{m+1}

(where

m

is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1)

P_0=(0,1),P_{m+1}=(n+1,n)

, and for each integer

i,1\le i\le m

, both

x

- and

y

- coordinates of

P_i

are integers lying in between

1

and

n

(

1

and

n

inclusive). 2) For each integer

i,0\le i\le m

P_iP_{i+1}

is parallel to the

x

-axis if

i

is even, and is parallel to the

y

-axis if

i

is odd. 3) For each pair

i,j

with

0\le i<j\le m

, line segments

P_iP_{i+1}

and

P_jP_{j+1}

share at most

1

point. Determine the maximum possible value that

m

can take.

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