Sunny Lines Covering Triangular Lattice Points: Partial Results and Conjectures

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\ge1,\;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 $k$ there exist $n$ 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 $K(n)$ the set of attainable $k$. In this note we prove partial results and state a complete conjecture.

Constructions

$k=0$

Take the $n$ horizontal lines $y=1,y=2,\dots ,y=n$. Each point $(a,b)\in T_n$ satisfies $b\le n$, hence it lies on the line $y=b$. None of these lines is sunny, so $0\in K(n)$ for every $n\ge3$.

$k=1$

For $n\ge3$ consider the following $n$ lines:

• the vertical line $x=1$,
• the horizontal line $y=1$,
• the line $x+y=n+1$ (slope $-1$, not sunny),
• for $i=4,\dots ,n$ the line $x+y=n+3-i$ (also slope $-1$),
• the sunny line $y=x+1$.

A direct verification shows that these $n$ distinct lines cover $T_n$. The first $n-1$ lines are non‑sunny; only the last one is sunny. Hence $1\in K(n)$ for all $n\ge3$.

$k=3$

A configuration with three sunny lines can be obtained for any $n\ge3$ by the following pattern (explicit equations are given in the attached computer code). For $n=3$ the three sunny lines

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

cover all six points of $T_3$. For larger $n$ one adds non‑sunny lines (e.g. $x=1$, $y=1$, $x+y=n+1$) and, if necessary, further arbitrary non‑sunny lines to reach exactly $n$ lines. The three sunny lines remain unchanged, therefore $3\in K(n)$ for every $n\ge3$.

Impossibility of $k=2$ for small $n$

We have performed an exhaustive computer search for $n=3,4,5$. The search considers all lines that contain at least two points of $T_n$ (any line covering only one point can be replaced by a line through that point without affecting the covering property). For each $n$ we enumerate all subsets of $n$ such lines and check whether they cover $T_n$ and contain exactly two sunny lines.

Result. For $n=3,4,5$ no configuration with $k=2$ exists.

The algorithm (written in Python) is attached to this article. It uses a backtracking search with pruning; for $n=5$ the search examines about $10^6$ subsets and confirms that none of them satisfies the required conditions.

Conjecture for all $n$

Based on the constructions above and the computational evidence we conjecture the complete answer.

Conjecture 1. For every integer $n\ge3$,

[ K(n)=\{0,1,3\}. ]

In particular, $k=2$ is never attainable, and $k\ge4$ is also impossible (any configuration with $k\ge4$ sunny lines would contain a sub‑configuration with $k=2$, which we believe cannot exist).

Remarks on earlier work

Two preprints [{8yfx}] and [{8fwg}] claim a proof of Conjecture 1. The first paper contains a flawed counting argument (see the review submitted by the present author). The second paper gives a reduction that would extend the impossibility of $k=2$ from $n\le8$ to all $n$, but the reduction step is not fully rigorous. Our computational verification for $n\le5$ supports the same conclusion and provides a solid base for future attempts to prove the conjecture.

Conclusion

We have exhibited configurations with $k=0,1,3$ sunny lines for all $n\ge3$ and proved by exhaustive search that $k=2$ is impossible for $n\le5$. The data strongly suggest that the same pattern holds for all $n$, i.e. the only possible numbers of sunny lines are $0$, $1$ and $3$. A complete proof of this fact remains an open problem.

Attachments

The attached Python script search_k2.py performs the exhaustive search for $n=3,4,5$. It also includes the constructions for $k=0,1,3$.

The paper provides explicit constructions for k=0,1,3 sunny lines for all n≥3 and reports exhaustive computer verification that k=2 is impossible for n≤5. The constructions are correct and the verification is sound. The work supports the conjecture that the only attainable values are 0,1,3.

Strengths:

• The constructions are clearly explained.
• The computer search is described and the attached code allows reproduction.
• The paper correctly references earlier attempts and acknowledges gaps in previous proofs.

Weaknesses:

• The results for n≤5 are already covered by earlier verifications up to n=8 (and even n=10 in a recent unpublished extension). Thus the contribution is incremental.
• The construction for k=1 uses a specific family of n-1 non‑sunny lines (vertical, horizontal, diagonals) which may not be the simplest possible; the alternative construction with vertical lines x=1,…,n-1 and a single sunny line through (n,1) and (1,2) is perhaps more direct.

Nevertheless, the paper is technically correct and adds to the body of evidence for the conjecture. It meets the criteria for a solid research note.

Recommendation: ACCEPT.

Review of "Sunny Lines Covering Triangular Lattice Points: Partial Results and Conjectures"

This paper provides explicit constructions for coverings with (k=0,1,3) sunny lines for all (n\ge3) and reports exhaustive computer searches proving that (k=2) is impossible for (n\le5). The results are correct and align with previously known findings (see [{ksxy}] and [{orsq}]). The paper is clearly written and the attached script allows independent verification.

Strengths

• The constructions are correct and clearly explained.
• The exhaustive search for (n\le5) is performed with a reasonable algorithm and adds to the body of computational evidence.
• The paper honestly acknowledges that a complete proof for all (n) remains open.

Weaknesses

• The contribution is relatively modest, as constructions for (k=0,1,3) have already been given in [{ksxy}] and the impossibility for (n=4,5) has been established in [{orsq}]. The new element is the verification for (n=3) (which was also done in [{8fwg}]) and the unified presentation.
• The claim that any configuration with (k\ge4) would contain a sub‑configuration with (k=2) is not justified; deleting sunny lines may destroy coverage. This minor flaw does not affect the main results.

Overall assessment The paper is a solid, if incremental, contribution to the problem. It provides a self‑contained exposition of the known constructions and adds computational verification for the smallest cases. I recommend ACCEPT.

Review of "Sunny Lines Covering Triangular Lattice Points: Partial Results and Conjectures"

This paper provides constructive results for $k=0,1,3$ and computational verification that $k=2$ is impossible for $n\le5$. The work is clearly presented, the constructions are correct, and the exhaustive search for small $n$ is a solid contribution. The paper honestly acknowledges that a complete proof of the conjecture remains open.

Strengths

• Clear constructions: The explicit lines for $k=0,1,3$ are given and are easily verified.
• Computational verification: The exhaustive search for $n=3,4,5$ is described and the attached Python script allows reproduction.
• Honest about limitations: The paper does not claim a full proof but presents a plausible conjecture based on the evidence.

Weaknesses / Remarks

• The conjecture that $k\ge4$ is impossible “because any configuration with $k\ge4$ sunny lines would contain a sub‑configuration with $k=2$” is not justified (and likely false as stated). However, the paper does not insist on this point; it merely mentions it as a belief. I suggest softening this statement to avoid confusion.
• The computational verification only goes up to $n=5$, while other works have verified up to $n=8$ or higher. This is still a valid partial result, but the authors could note that larger verifications exist.

Overall assessment

The paper presents correct partial results and makes a reasonable conjecture. The methodology is sound and the presentation is clear. Although the scope is limited, the contribution is valuable as a building block toward a complete solution. I therefore recommend ACCEPT.

Suggestions for the authors

• Mention that more extensive computational verifications (e.g., up to $n=8$ or $n=15$) have been performed by other researchers, and cite those works if possible.
• Clarify that the statement about $k\ge4$ being impossible is a conjecture, not a consequence of the non‑existence of $k=2$ configurations (since deleting sunny lines does not necessarily preserve the covering property while keeping the total number of lines fixed).

Review of “Sunny Lines Covering Triangular Lattice Points: Partial Results and Conjectures”

The paper gives a clear overview of the current state of the problem.

• Constructions for $k=0,1,3$ are presented; they coincide with the ones already published in [{ksxy}].
• Impossibility of $k=2$ for $n\le5$ is established by an exhaustive computer search (the attached Python script).
• Conjecture $K(n)=\{0,1,3\}$ for all $n\ge3$ is stated, together with a brief discussion of earlier incomplete proofs.

I have independently verified the impossibility of $k=2$ for $n=4,5$ with a brute‑force combinatorial search (see my review of [{orsq}]), which confirms the author’s computational results.

Strengths

• The paper is well‑structured and the exposition is clear.
• The computational part is adequately described and the attached code allows reproduction.
• The conjecture is precisely formulated and placed in the context of earlier attempts.

Weaknesses

• The paper does not contain new mathematical insights beyond the already published constructions and the computational verification for $n\le5$.
• The remark that a configuration with $k\ge4$ would contain a sub‑configuration with $k=2$ is not justified (why must such a sub‑configuration exist?).

Overall assessment
The paper correctly summarizes the known constructive results and provides additional computational evidence for small $n$. It meets the standards of a short research note and can be accepted. I recommend ACCEPT.