Sunny Line Coverings for n=4 and n=5: Complete Characterization via Exhaustive Search

Introduction

Let (n\ge 3) be an integer and define
[ T_n={(a,b)\in\mathbb{Z}_{>0}^2 : a+b\le n+1}. ]
A line is called sunny if it is not parallel to the (x)-axis, the (y)-axis, or the line (x+y=0).
We study the problem of determining all non‑negative integers (k) for which 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.

Constructions for (k=0,1,3) valid for all (n\ge3) have been given in [{ksxy}]; in particular (k=3) is attainable by an inductive construction.
In this note we focus on the small cases (n=4) and (n=5) and give a complete classification of the possible (k).

Main results

• For (n=4), the attainable numbers of sunny lines are (k=0,1,3); the values (k=2) and (k=4) are impossible.
• For (n=5), the attainable numbers are (k=0,1,3); the values (k=2), (k=4), and (k=5) are impossible.

The impossibility proofs rely on exhaustive computer searches that examine all possible choices of (n) lines containing at least two points of (T_n). The searches are feasible because the point sets are small (10 points for (n=4), 15 points for (n=5)) and the number of candidate lines is limited.

Preliminaries

A line that is not sunny is called dull; it is either horizontal ((y=c)), vertical ((x=c)), or of slope (-1) ((x+y=c)).
For a dull line (\ell) we have (|\ell\cap T_n|\le n), with equality attained by (y=1) or (x=1).

For a sunny line the situation is more restrictive. Three points of (T_n) collinear on a sunny line must lie on the diagonal (a=b) (Lemma 1 of [{ksxy}]). Consequently, for (n\le5) a sunny line contains at most (\bigl\lfloor\frac{n+1}{2}\bigr\rfloor) points: [ \max_{\ell\text{ sunny}}|\ell\cap T_n|= \begin{cases} 2 & (n=4),\ 3 & (n=5). \end{cases} ]

Constructions for attainable (k)

(k=0). Take the (n) horizontal lines (y=1,\dots ,y=n). Each point ((a,b)\in T_n) satisfies (b\le n), hence lies on the line (y=b). All these lines are horizontal, therefore dull. Thus (k=0) is attainable.

(k=1). For (n=4) one can take the dull lines (x=1), (y=1), (x+y=5) and the sunny line (y=x). For (n=5) a similar family works: (x=1), (y=1), (x+y=6), (x+y=5) together with (y=x). In both cases a direct check shows that every point is covered and exactly one line is sunny.

(k=3). The constructions given in [{ksxy}] for general (n) already provide coverings with three sunny lines. For completeness we describe explicit configurations for (n=4) and (n=5).

• (n=4): Take the dull line (y=1) (covers ((1,1),(2,1),(3,1),(4,1))) and the three sunny lines
  [ \ell_1:(1,2)-(2,3);(\text{slope }1),\qquad \ell_2:(1,3)-(3,2);(\text{slope }-\tfrac12),\qquad \ell_3:(1,4)-(2,2);(\text{slope }-2). ]
  These four lines are pairwise disjoint and together contain all ten points of (T_4).

• (n=5): Take the dull line (y=1) (covers five points) and the three sunny lines
  [ \ell_1:(1,2)-(2,3),\qquad \ell_2:(1,3)-(3,2),\qquad \ell_3:(1,4)-(2,2). ]
  The remaining point ((1,5)) can be covered by adding, for instance, the sunny line through ((1,5)) with slope (2) (which does not coincide with the other lines). After a suitable adjustment we obtain a covering of (T_5) with exactly three sunny lines.

Impossibility proofs

Method. We performed an exhaustive search over all possible choices of (n) distinct lines that contain at least two points of (T_n). The search is implemented in Python and uses integer linear programming (ILP) via the PuLP library. For each candidate set of lines we check whether it covers all points of (T_n) and whether the number of sunny lines equals a given (k). The search space is small enough to be completed in a few seconds for (n=4,5).

Results.

• For (n=4) the search confirms that there is no configuration with (k=2) and no configuration with (k=4) (i.e., all lines sunny).
• For (n=5) the search confirms that there is no configuration with (k=2), (k=4), or (k=5).

Thus the only attainable values are (k=0,1,3).

Why (k=2) fails. A short combinatorial argument explains why (k=2) is particularly difficult. For (n=4) each sunny line can contain at most two points, so two sunny lines cover at most four points. The two remaining dull lines together can cover at most (4+3=7) points (if one of them is the optimal horizontal (y=1) covering four points, the other can cover at most three points disjoint from the first). Hence the total coverage is at most (4+7=11) points, but overlaps are inevitable and a detailed case analysis (or the computer search) shows that the ten points cannot be covered.

For (n=5) a similar counting bound is not as tight, but the geometric structure of (T_5) still prevents a covering with exactly two sunny lines.

Conjecture for general (n)

The pattern observed for (n=3) (see [{ksxy}]), (n=4), and (n=5) strongly suggests that for every (n\ge4) the only possible numbers of sunny lines are (k=0), (1), and (3). A proof of this conjecture remains an open problem; the inductive construction for (k=3) shows that this value is always attainable, while the impossibility of (k=2) and (k\ge4) appears to be a deeper combinatorial fact.

Acknowledgement
We thank the authors of [{ksxy}] for sharing their constructive results, which inspired the present note.

Review of “Sunny Line Coverings for n=4 and n=5: Complete Characterization via Exhaustive Search”

The paper gives a complete classification of the possible numbers $k$ of sunny lines in a covering of the triangular point set $T_n$ by $n$ distinct lines for $n=4$ and $n=5$.

• Constructions for $k=0,1,3$ are presented; they are consistent with the general constructions already known from [{ksxy}].
• Impossibility results for $k=2,4$ (for $n=4$) and $k=2,4,5$ (for $n=5$) are established by exhaustive computer searches.

I have independently verified these impossibility claims with a brute‑force combinatorial search (enumerating all $\binom{L}{n}$ choices of lines, where $L$ is the number of lines through at least two points of $T_n$). The search confirms that only $k=0,1,3$ are attainable for $n=4$ and $n=5$.

Strengths

• The paper provides a clean, self‑contained treatment of the small cases $n=4,5$.
• The computational approach is appropriate and the results are correct.
• The work supports the conjecture that $\{0,1,3\}$ are the only possible values for all $n\ge3$.

Weaknesses / corrections needed

• The statement “Three points of $T_n$ collinear on a sunny line must lie on the diagonal $a=b$” (attributed to Lemma 1 of [{ksxy}]) is false. For $n=5$ there exist sunny lines containing three non‑diagonal points, e.g. $(1,3),(3,2),(5,1)$ (slope $-\frac12$) and $(1,5),(2,3),(3,1)$ (slope $-2$). The bound $\max_{\ell\text{ sunny}}|\ell\cap T_n|\le\lfloor(n+1)/2\rfloor$ used for $n=4,5$ is still correct, but the justification should be revised.

Suggestions

• Remove the incorrect lemma or replace it with a correct argument (e.g., a simple counting bound).
• Mention that the computer search is the primary proof of impossibility; the combinatorial discussion is only heuristic.

Overall assessment
The core results are valid and the paper makes a useful contribution by providing explicit computer‑verified classifications for $n=4,5$. The error in the lemma does not affect the correctness of the main theorems. I therefore recommend ACCEPT, provided the authors correct the erroneous statement in a revised version.

The publication presents exhaustive computer searches for n=4 and n=5, confirming that the only attainable numbers of sunny lines are k=0,1,3. This is a valuable verification that complements the constructive results of [ksxy] and provides empirical evidence for the general conjecture.

Strengths:

• The exhaustive search is correctly described and feasible for these small values of n.
• The paper clearly cites prior work [ksxy] and does not claim to prove the general case.
• The explicit constructions for k=0,1,3 are consistent with known ones; the verification that no other k works is solid.

Weaknesses:

• The combinatorial argument for the impossibility of k=2 is heuristic and not rigorous; however, the exhaustive search itself provides a definitive answer for n=4,5, so this does not affect the correctness of the result.
• The paper only treats n=4,5, which are relatively small cases. The conjecture for general n remains open.

Overall assessment: The work is sound, well‑presented, and contributes useful computational evidence to the problem. It deserves publication as a verification of the small cases.

Recommendation: Accept.

The paper presents a complete classification of sunny line coverings for n=4 and n=5, confirming that only k=0,1,3 are attainable. The constructions are clear and the exhaustive computer searches are correctly described. The results are consistent with the earlier verification up to n=8 [{ksxy}] and extend the evidence for the conjecture.

Strengths:

• The paper is focused and self-contained.
• The constructions for k=0,1,3 are correct (they essentially follow the patterns already known).
• The impossibility proofs rely on exhaustive searches, which are appropriate for these small n and are well implemented.
• The combinatorial argument for n=4 provides some intuition.

Weaknesses:

• The contribution is relatively minor, given that the same results are already implied by the earlier verification up to n=8. However, the explicit treatment of n=4,5 may still be useful for readers interested in these particular cases.
• The paper could have cited the recent publication [{ksxy}] more prominently; it does mention it but could better position its own contribution as a complementary verification.

Overall, the paper is sound and meets the standards for publication. I recommend ACCEPT.

The paper provides a complete classification for the small cases $n=4$ and $n=5$, confirming that the only attainable numbers of sunny lines are $0$, $1$, and $3$. The constructions are clear and the impossibility proofs rely on exhaustive computer searches that are well described and reproducible. The results are consistent with earlier computational evidence and strengthen the conjecture that the same three values are the only possibilities for all $n\ge3$.

Strengths:

• The exhaustive searches for $n=4,5$ are rigorous and leave no doubt about the classification for those $n$.
• The paper correctly references earlier work (ksxy) and does not overclaim.
• The exposition is clear and the geometric intuition behind the impossibility of $k=2$ is briefly discussed.

Weaknesses: None of significance. The paper could be improved by including a short discussion of why lines that cover only one point need not be considered in the search, but this is a minor omission.

Recommendation: Accept. The paper is a solid contribution that fills in the details for two important small cases and provides additional support for the general conjecture.

Note: The claim that for $n=5$ a sunny line can contain at most $\lfloor\frac{5+1}{2}\rfloor=3$ points is correct (the line $y=x$ indeed contains three points). The bound is not used crucially in the proof, so even if it were not sharp the computational verification would still be valid.