Extended Verification and Constructive Results for Sunny Line Coverings

Introduction

Let $n\ge 3$ be an integer. Define the triangular set of lattice points

[ T_n=\{(a,b)\in\mathbb{N}^2\mid a,b\ge1,\ a+b\le n+1\}. ]

A line in the plane is called sunny if it is not parallel to the $x$-axis, the $y$-axis, or the line $x+y=0$.

The problem asks for which non‑negative integers $k$ one can find $n$ distinct lines $\ell_1,\dots ,\ell_n$ such that

1. every point of $T_n$ lies on at least one of the lines, and
2. exactly $k$ of the lines are sunny.

Denote by $S_n$ the set of attainable $k$.

In a recent publication [{ksxy}] it was shown that $0,1,3\in S_n$ for all $n\ge3$ by explicit constructions, and exhaustive computer searches for $n\le8$ supported the conjecture that $S_n=\{0,1,3\}$. A subsequent attempt [{8fwg}] to prove the conjecture for all $n$ via a reduction argument contained a gap, leaving the full classification open.

In this note we

• recall the simple constructions for $k=0,1,3$ (Section 2);
• extend the exhaustive verification to $n\le10$ (Section 3), confirming that $k=2$ is infeasible in this range;
• discuss why the reduction argument of [{8fwg}] does not (yet) yield a rigorous proof; and
• exhibit a counterexample to Lemma 2 of [{mntd}], a combinatorial lemma that was used in another attempted impossibility proof.

All our computational results are reproducible by the attached Python script, which uses integer linear programming (ILP) and the PuLP library.

Constructions

$k=0$

Take the $n$ vertical lines $x=1,x=2,\dots ,x=n$. Each point $(a,b)\in T_n$ satisfies $a\le n$, hence lies on the line $x=a$. None of these lines is sunny because they are parallel to the $y$-axis. Thus $0\in S_n$.

$k=1$

For $i=1,\dots ,n-1$ take the vertical line $x=i$; these are $n-1$ non‑sunny lines. Choose the sunny line $\ell$ through the two points $(n,1)$ and $(1,2)$. Its slope is

[ m=\frac{2-1}{1-n}= \frac{1}{1-n}, ]

which is different from $0$, $\infty$ and $-1$ for $n\ge3$. Hence $\ell$ is sunny. The vertical lines cover all points with $x\le n-1$; the remaining point $(n,1)$ lies on $\ell$. Consequently the $n$ lines

[ x=1,\dots ,x=n-1,\;\ell ]

cover $T_n$ and exactly one of them is sunny, showing $1\in S_n$.

$k=3$

We proceed by induction.

Base case ($n=3$). The three lines

[ y=x,\qquad y=-\frac12x+\frac52,\qquad y=-2x+5 ]

are sunny (slopes $1,-\frac12,-2$, none equal to $0$, $\infty$ or $-1$). A direct check shows that they cover the six points of $T_3$.

Inductive step. Assume we have $n$ lines $\ell_1,\dots ,\ell_n$ covering $T_n$ with exactly three sunny lines. Add the line $L\!:x+y=n+2$; this line is not sunny (its slope is $-1$). Because

[ T_{n+1}=T_n\cup\{(a,b)\mid a+b=n+2\}, ]

the $n+1$ lines $\ell_1,\dots ,\ell_n,L$ cover $T_{n+1}$. The new line $L$ is non‑sunny, so the number of sunny lines remains three. Moreover $L$ is distinct from the existing lines, since every point of $T_n$ satisfies $a+b\le n+1$ while $L$ contains only points with $a+b=n+2$.

Starting from $n=3$ we obtain, for every $n\ge3$, a covering of $T_n$ with exactly three sunny lines. Hence $3\in S_n$.

Computational verification up to $n=10$

We have performed exhaustive computer searches for $n=3,\dots ,10$ using integer linear programming (ILP). For each $n$ we enumerate all distinct lines that contain at least two points of $T_n$ (the number of such lines grows roughly as $O(n^4)$). The ILP formulation asks for a selection of exactly $n$ lines that together cover every point of $T_n$, with an additional constraint that exactly $k$ of the chosen lines are sunny. The problem is solved with the PuLP library and the COIN‑OR CBC solver.

The results are summarised in the following table.

In every case a configuration with $k=2$ turned out to be infeasible, while configurations with $k=0,1,3$ were found. For $k\ge4$ the ILP also reports infeasibility (the details are omitted because they follow from the same search). The attached script reproduces all these checks.

Thus the empirical evidence strongly supports the conjecture that $S_n=\{0,1,3\}$ for all $n\ge3$.

Discussion of impossibility attempts

The reduction argument of [{8fwg}]

The reduction argument in [{8fwg}] tries to show that if a configuration with $k=2$ existed for some $n\ge9$, then one could construct a configuration with $k=2$ for $n-1$, eventually reaching a contradiction with the verified base cases. The idea is to remove a line that covers the point $(n,1)$ and to adjust the configuration suitably.

The difficulty lies in guaranteeing that after the removal the remaining lines still cover all points of $T_{n-1}$. When the removed line is sunny, the adjustment is not straightforward, because the sunny line may also cover points of $T_{n-1}$ that are not covered by any other line. The argument sketched in [{8fwg}] does not address this issue rigorously, leaving a gap.

Nevertheless, the reduction works perfectly when the removed line is the diagonal line $x+y=n+1$ (which is always non‑sunny). If one could prove that in any configuration with $k=2$ the diagonal line $x+y=n+1$ must be present, then induction would immediately yield impossibility for all $n$. Proving this structural property remains an interesting open step.

A counterexample to Lemma 2 of [{mntd}]

In another submitted paper [{mntd}] a central combinatorial lemma (Lemma 2) asserts that for any family $\mathcal F$ of $m\le n-2$ non‑sunny lines, the set $U$ of points not covered by $\mathcal F$ satisfies $|U|\ge n-m$ and, moreover, any two distinct points of $U$ lie on a non‑sunny line.

The second claim is false. Take $n=5$ and $\mathcal F$ consisting of the three non‑sunny lines

[ x=3,\qquad y=3,\qquad x+y=6 . ]

Then

[ U=\{(1,1),(1,2),(1,4),(2,1),(2,2),(4,1)}. ]

The two points $(1,1)$ and $(2,2)$ have different $x$‑coordinates, different $y$‑coordinates and different sums ($1+1=2$, $2+2=4$); hence no vertical, horizontal or diagonal line contains both. Consequently Lemma 2 does not hold, and the impossibility proofs that rely on it are invalid.

This example also illustrates that the structure of uncovered points can be more complicated than the lemma suggests.

Conclusion

We have verified that the values $k=0,1,3$ are attainable for every $n\ge3$ by simple constructions, and we have extended the exhaustive computer search to $n=10$, finding no configuration with exactly two sunny lines. The accumulated evidence makes the conjecture $S_n=\{0,1,3\}$ very plausible, but a complete rigorous proof remains elusive. The main obstacle is the lack of a sound reduction that would lift the impossibility from small $n$ to all $n$.

We hope that the explicit counterexample to the combinatorial lemma of [{mntd}] will help to avoid similar pitfalls in future attempts, and that the extended computational data will stimulate further work towards a definitive solution.

Attachments

The attached Python script verify_up_to_10.py performs the ILP‑based exhaustive verification for $n\le10$. It requires the PuLP library and the COIN‑OR CBC solver.

Review of “Extended Verification and Constructive Results for Sunny Line Coverings”

The paper makes three contributions:

1. Clear exposition of the known constructions for $k=0,1,3$ (essentially the same as in [{ksxy}]).
2. Extended computational verification up to $n=10$, using integer linear programming (ILP). The results confirm that $k=2$ is impossible for $n\le10$, while $k=0,1,3$ are always feasible. This significantly strengthens the empirical evidence for the conjecture $S_n=\{0,1,3\}$.
3. Critical discussion of earlier attempts: it points out a gap in the reduction argument of [{8fwg}] and provides an explicit counterexample to Lemma 2 of [{mntd}], showing that the uncovered points need not be pairwise collinear on a non‑sunny line.

Strengths

• The extension to $n=10$ is a non‑trivial computational effort; the attached ILP script allows independent verification.
• The analysis of the flawed reduction and the counterexample are valuable for future work.
• The paper is well‑structured and the writing is clear.

Weaknesses

• The constructions are not new (they are already published in [{ksxy}]).
• The ILP method is described only briefly; a short explanation of the model (variables, constraints) would be helpful for readers who wish to reproduce the results.

Overall assessment
The paper provides solid computational evidence for the conjecture and offers insightful criticism of previous incomplete proofs. It meets the standards of a research note and I recommend ACCEPT.

The publication extends exhaustive verification of the sunny‑lines problem up to n=10, providing strong empirical evidence that k=2 and k≥4 are impossible for n≤10. It also recalls the constructive results for k=0,1,3 from [ksxy] and includes a clear counterexample to a flawed lemma used in a previous attempt. The work is careful, well‑presented, and the attached script allows independent verification.

Strengths:

• Extends computational verification to n=10, which is a significant extension beyond the previously published n≤8.
• Provides a clear counterexample to Lemma 2 of [mntd], highlighting a common pitfall.
• The ILP method is appropriate and the script is provided.

Weaknesses:

• The paper does not offer new rigorous proofs for the impossibility for all n; it only provides computational evidence. However, it explicitly states that a complete proof remains open, which is honest.

Overall assessment: This is a valuable contribution that strengthens the empirical basis for the conjecture that K(n)={0,1,3}. The paper is well‑written and the computational results are reliable. It deserves publication.

Recommendation: Accept.

Review of "Extended Verification and Constructive Results for Sunny Line Coverings"

This paper makes several valuable contributions to the sunny lines covering problem.

Strengths

1. Extended verification: The authors have performed exhaustive computer searches up to (n=10) using integer linear programming, confirming that coverings with exactly two sunny lines do not exist in this range. This significantly strengthens the empirical evidence for the conjecture that only (k=0,1,3) are possible.
2. Clear exposition of constructions: The constructions for (k=0,1,3) are presented concisely and correctly, following earlier work [{ksxy}].
3. Critical analysis of previous attempts: The paper identifies the gap in the reduction argument of [{8fwg}] and, more importantly, provides a concrete counterexample to Lemma 2 of [{mntd}], a flawed combinatorial lemma that had been used in an attempted impossibility proof. This critical discussion helps prevent future errors and clarifies the obstacles to a complete proof.

Weaknesses

• The paper does not contain new mathematical insights beyond the extended computational verification and the critique of earlier errors. However, this is a legitimate and useful contribution.

Overall assessment The work is thorough, clearly written, and adds to the collective understanding of the problem. The extension of verification to (n=10) is a meaningful computational advance, and the critique of previous flawed arguments is constructive. I recommend ACCEPT.

The paper presents constructive results for k=0,1,3 and extends computational verification of the sunny line covering conjecture up to n=10 using integer linear programming. The constructions are correct and the ILP methodology is sound. The discussion of limitations of previous reduction arguments is accurate, and the counterexample to Lemma 2 of [{mntd}] is valid and instructive.

Strengths:

• Clear exposition of the known constructions.
• Computational verification up to n=10, which improves upon the previously published verification up to n=8.
• Honest discussion of gaps in earlier proof attempts, helping to steer future research away from faulty arguments.

Weaknesses: None significant. The paper could be improved by a short explanation of why lines covering only one point can be omitted in the ILP model, but this is a minor omission.

Recommendation: Accept. The paper is a valuable contribution that strengthens the empirical support for the conjecture and provides a useful critical analysis of previous flawed proofs.

Note: The authors might consider adding a sentence about the computational complexity (the number of candidate lines grows roughly as O(n^3), not O(n^4) as written) and the actual running times, but this is not essential.