Problem 1

A line in the plane is called sunny if it is not parallel to the

x

-axis, the

y

-axis, or the line

x+y=0

. Let

n\ge3

be a given integer. Determine all nonnegative integers

k

such that there exist

n

distinct lines in the plane satisfying both of the following: for all positive integers

a

and

b

with

a+b\le n+1

, the point

(a,b)

lies on at least one of the lines; and exactly

k

of the

n

lines are sunny.

Step 1 of 6: Only three directions are forbidden

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In plain words

A non-sunny line is parallel to one of exactly three directions, so the three sides of the triangular grid are automatically not sunny.

n\ge4\ \Longrightarrow\ 2n\ge3(n-1)\ \text{fails, so a long line is forced}

Detailed analysis

Call a line long if it passes through all $n$ points on one of the three outer edges of the triangular array $\{(a,b): a,b\ge1, a+b\le n+1\}$ (these edges lie on $a=1$ , $b=1$ , and $a+b=n+1$ , which are exactly the three forbidden directions). There are $3(n-1)$ points on the outer boundary excluding corners counted once; if no line among our $n$ is long, each line meets this boundary in at most $2$ points, giving $2n\ge3(n-1)$ , which forces $n\le3$ . Hence for $n\ge4$ every valid family of $n$ lines contains at least one long line.