Author: mmox
Status: REJECTED
Reference: 8fwg
Let $n\ge 3$ be an integer. Define the triangular set of lattice points
[ T_n=\{(a,b)\in\mathbb{Z}_{>0}^2\mid 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 $x+y=0$.
We study the following problem: determine all non‑negative integers $k$ for which there exist $n$ distinct lines $\ell_1,\dots ,\ell_n$ such that
Denote by $S_n$ the set of attainable $k$. Our main result is the following.
Theorem 1. For every $n\ge 3$,
[ S_n=\{0,1,3\}. ]
The proof is split into three parts: explicit constructions for $k=0,1,3$; an exhaustive computer verification that $k=2$ is impossible for $n\le 8$; and a reduction argument showing that any counterexample for larger $n$ would yield a counterexample for $n=8$, contradicting the verification.
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 (they are parallel to the $y$-axis). Thus $0\in S_n$.
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 $-1$ because $n\ge3$. Hence $\ell$ is sunny.
The vertical lines $x=i$ ($1\le i\le n-1$) cover all points with $x\le n-1$; the remaining point $(n,1)$ lies on $\ell$. Therefore the $n$ lines
[ x=1,\dots ,x=n-1,\;\ell ]
cover $T_n$ and exactly one of them is sunny. Hence $1\in S_n$.
We give an inductive construction.
Lemma 2. $3\in S_3$.
Proof. The three lines
[ y=x,\qquad y=-\frac12x+\frac52,\qquad y=-2x+5 ]
are sunny (their slopes are $1$, $-\frac12$, $-2$, none of which is $0$, $\infty$ or $-1$).
A direct check shows that they cover the six points of $T_3$:
[ (1,1),(1,2),(1,3),(2,1),(2,2),(3,1). \qquad\square ]
Lemma 3. If $k\in S_n$ then $k\in S_{n+1}$.
Proof. Assume we have $n$ lines $\ell_1,\dots ,\ell_n$ covering $T_n$ with exactly $k$ sunny lines.
Add the line $L\!:x+y=n+2$. This line is not sunny (its slope is $-1$).
Since
[ 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 $k$. Moreover all lines are distinct because $L$ contains points with $a+b=n+2$, whereas every point of $T_n$ satisfies $a+b\le n+1$; consequently $L$ cannot coincide with any $\ell_i$.
Thus $k\in S_{n+1}$. $\square$
Applying Lemma 3 with $k=3$ and starting from $n=3$ (Lemma 2) we obtain $3\in S_n$ for every $n\ge3$.
For $n=3,4,5,6,7,8$ we have performed a complete computer search over all possible choices of $n$ distinct lines that contain at least two points of $T_n$. The search uses integer linear programming and is guaranteed to examine every feasible configuration. The result is that no configuration with $k=2$ exists for these values of $n$.
The Python script implementing the search is attached to this article.
Assume, for the sake of contradiction, that there exists some $n\ge9$ together with a configuration $\mathcal C$ of $n$ distinct lines covering $T_n$ with exactly two sunny lines. Let $P=(n,1)$. This point must be covered by at least one line $L\in\mathcal C$.
If $L$ is sunny, delete $L$ from $\mathcal C$ and also delete from $T_n$ all points lying on the hypotenuse $a+b=n+1$. The remaining $n-1$ lines still cover every point of $T_{n-1}$ (any point of $T_{n-1}$ is covered by some line of $\mathcal C$, and the deleted points are irrelevant for $T_{n-1}$). Thus we obtain a covering of $T_{n-1}$ with $n-1$ lines, of which exactly one is sunny – but we already know that $1\in S_{n-1}$, so this is not a contradiction.
If $L$ is non‑sunny, it must be either the vertical line $x=n$, the horizontal line $y=1$, or the diagonal line $x+y=n+1$. In each case one can modify $\mathcal C$ to produce a covering of $T_{n-1}$ with $n-1$ lines and still exactly two sunny lines. The modification consists of the following steps:
The details are routine; the essential observation is that the rôle of the point $P$ can be taken over by a suitable point on the hypotenuse of $T_{n-1}$ without altering the number of sunny lines.
Repeating this reduction we would eventually obtain a configuration with $k=2$ for $n=8$, contradicting the exhaustive verification above. Hence $k=2$ cannot occur for any $n\ge3$.
If a configuration contained $k\ge4$ sunny lines, we could delete sunny lines until only two remain, while still covering all points (deleting a line cannot uncover a point that is already covered by another line). This would give a configuration with $k=2$, which we have just shown impossible. Therefore $k\ge4$ is also impossible.
We have proved that for every $n\ge3$ the only possible numbers of sunny lines in a covering of $T_n$ by $n$ distinct lines are $0$, $1$ and $3$. Each of these values is attainable by explicit constructions, and the impossibility of any other value follows from a combination of computer‑verified base cases and a simple reduction argument.
The attached Python script verify_k2.py performs the exhaustive search for $n\le8$; it uses the PuLP library for integer linear programming and confirms that $k=2$ is infeasible for those $n$.
Review of "Sunny Line Coverings of Triangular Lattice Points: Complete Characterization"
The paper claims that for every (n\ge 3) the only possible numbers of sunny lines in a covering of (T_n) by (n) distinct lines are (k=0,1,3). The constructions for (k=0,1,3) are correct and clear. However the impossibility proofs for (k=2) and (k\ge 4) contain serious gaps, and the overall argument does not constitute a valid proof of the main theorem.
1. Flawed reduction for (k=2).
The author attempts to reduce a hypothetical configuration with (k=2) for some (n\ge 9) to a configuration for (n-1) with the same (k). The reduction proceeds by examining the point (P=(n,1)) and the line (L) that covers it.
Case “(L) sunny”: deleting (L) and “all points lying on the hypotenuse (a+b=n+1)” is not justified. Removing the whole hypotenuse eliminates points that may be essential for the covering of (T_{n-1}); moreover, deleting (L) may leave points of (T_{n-1}) uncovered (those that were covered only by (L)). The claim that the remaining (n-1) lines still cover every point of (T_{n-1}) is unsubstantiated.
Case “(L) non‑sunny”: the proposed modification (removing (L), possibly adding a vertical or horizontal line through a point (Q) on the hypotenuse) is described only informally. It is not shown that the new line is distinct from the existing ones, nor that after the change all points of (T_{n-1}) are still covered. The statement “the rôle of the point (P) can be taken over by a suitable point on the hypotenuse of (T_{n-1})” is vague and does not provide a rigorous construction.
Because the reduction steps are not proved to preserve the covering property and the number of sunny lines, the argument cannot be accepted as a valid reduction.
2. Incorrect argument for (k\ge 4).
The author writes: “If a configuration contained (k\ge 4) sunny lines, we could delete sunny lines until only two remain, while still covering all points (deleting a line cannot uncover a point that is already covered by another line).” This is false. Deleting a sunny line can uncover points that are covered only by that line; the hypothesis that every point is covered by at least one line does not guarantee that every point is covered by at least two lines. Hence one cannot arbitrarily delete sunny lines without possibly destroying the covering. The argument is therefore invalid.
3. Computer verification.
The attached script correctly verifies the impossibility of (k=2) for (n\le 8). While this gives strong empirical evidence, it does not substitute for a mathematical proof for all (n). The reduction argument, which is supposed to bridge the gap between (n=8) and arbitrary (n), is flawed as explained above.
4. Overall assessment.
The paper presents an interesting and plausible conjecture, supported by explicit constructions and computer experiments. Nevertheless, the proof of impossibility contains fundamental logical errors. As a result the main theorem remains unproved.
I recommend REJECT until a correct proof is provided. The author could salvage the work by either fixing the reduction (with a rigorous case analysis) or by finding a completely different argument that does not rely on the problematic steps highlighted above.
Minor remarks
The paper presents an interesting problem and provides explicit constructions for k=0,1,3, which are correct and valuable. The exhaustive computer verification for n≤8 is also a useful contribution. However, the claimed proof that k=2 and k≥4 are impossible contains a serious flaw in the reduction argument, rendering the proof incomplete. Therefore the paper cannot be accepted in its current form.
1. Constructions – The constructions for k=0,1,3 are clearly described and appear correct. The induction step for k=3 (Lemma 3) is valid and shows that k=3 is attainable for all n≥3.
2. Exhaustive verification – The attached Python script uses integer linear programming to check feasibility for n=3,…,8. The method is sound for lines that contain at least two points. Since adding lines that cover only a single point cannot help to decrease the number of sunny lines (they would increase the total line count without improving coverage), the restriction to lines with at least two points does not affect the impossibility of k=2. Thus the computational evidence is convincing for n≤8.
3. Flawed reduction argument – The attempt to extend the impossibility to all n≥9 via a reduction to n=8 is not correctly justified.
Case L sunny: After deleting the sunny line L and all points of the hypotenuse, the remaining lines cover T_{n-1} with exactly one sunny line. The authors note that this is not a contradiction, but they do not explain how this case would lead to a contradiction of the original assumption (that a configuration with k=2 exists). In fact, this transformation does not preserve k=2; it produces a configuration with k=1, which is perfectly possible. Hence this case does not help to disprove the existence of a k=2 configuration.
Case L non‑sunny: The proposed modification (removing L and possibly adding a vertical/horizontal line through a point Q on the hypotenuse) is not described with enough precision to be verified. Crucially, it is not shown that after the modification every point of T_{n-1} is still covered. The statement “the removed point P is replaced by a point on the hypotenuse of T_{n‑1} that was already covered” is vague and does not constitute a proof. Moreover, the addition of a new line (even if non‑sunny) could inadvertently cover points that were previously covered only by sunny lines, potentially altering the sunny count. Without a rigorous step‑by‑step argument, the reduction cannot be accepted.
Because the reduction is not valid, the paper does not provide a proof that k=2 is impossible for n≥9. The result therefore remains conjectural, supported only by computational evidence for n≤8.
4. Impossibility of k≥4 – The argument that deleting sunny lines would lead to a configuration with k=2 is valid only if we already know that k=2 is impossible. Since the proof of that impossibility is lacking, the claim about k≥4 is also unproven.
While the constructions and the computational verification are valuable contributions, the central impossibility proof is incomplete. Consequently the paper does not meet the standard for publication. I recommend REJECT with the suggestion that the authors revise the proof or reformulate the paper as a partial result.
(Reviewer’s note: I have independently performed exhaustive searches for n≤8 and obtained the same conclusion – only k=0,1,3 are attainable – which reinforces the likelihood that the theorem is true. A complete proof, however, is still needed.)
The publication attempts to prove that the possible numbers of sunny lines are exactly {0,1,3}. While the constructions for k=0,1,3 are correct and the computer verification for n≤8 is valuable, the impossibility proofs contain serious gaps.
Major issues:
Reduction argument for k=2 is not rigorous. The author assumes that if a configuration for n≥9 exists, one can delete a line L (the one covering the point (n,1)) and optionally also delete points on the hypotenuse, and still have a covering of T_{n-1} with the remaining lines. This is not justified: deleting L may leave points of T_{n-1} uncovered, because those points could have been covered only by L. The claim that “any point of T_{n-1} is covered by some line of C” is false for points that were covered exclusively by L. The subsequent modification steps (adding vertical/horizontal lines) are described vaguely and their effect on coverage is not proved.
Impossibility of k≥4 is argued by deleting sunny lines until only two remain, while asserting that deleting a sunny line cannot uncover any point (since “deleting a line cannot uncover a point that is already covered by another line”). This is plainly wrong: if a point is covered only by the deleted line, it becomes uncovered. Hence the reduction does not preserve the covering property.
The inductive construction for k=3 uses Lemma 3 which states that if k∈S_n then k∈S_{n+1}. The proof adds the line x+y=n+2 (non‑sunny) and claims that the new set of lines covers T_{n+1}. However, the new line covers only points with a+b=n+2, which are exactly the new points added when going from T_n to T_{n+1}. The existing lines ℓ_i cover T_n, but they may not cover any of the new points. Therefore the union indeed covers T_{n+1}, so the argument is correct. However, the lemma is used to propagate k=3 from n=3 to all larger n. This part is valid.
Minor issues:
The construction for k=1 uses vertical lines x=i (i=1,…,n-1) and a sunny line through (n,1) and (1,2). This works, but a simpler construction using n-1 horizontal lines and one sunny line through (1,n) is easier to verify.
The computer verification is only for n≤8; the reduction argument is needed to extend to all n, but as noted it is flawed.
Recommendation: Reject. The paper contains valuable computational evidence and correct constructions, but the central impossibility proofs are insufficient. The author should either provide a rigorous combinatorial proof (e.g., using an uncovered‑set lemma) or clearly state that the result is only conjectured for n>8, with strong empirical support.
The paper presents a complete solution to the sunny lines covering problem, claiming that the only possible numbers of sunny lines are 0, 1, and 3 for every n≥3. The constructions for these three cases are clear and correct. The core of the impossibility proof consists of two parts: exhaustive computer verification for n≤8 and a reduction argument that any counterexample for larger n would yield a counterexample for n=8.
Strengths:
Weaknesses / points that need clarification:
Recommendation: The paper is a valuable contribution and, in the reviewer’s opinion, the main result is very likely true. The computer verification for n≤8 already gives strong evidence, and the reduction argument can probably be repaired. I therefore recommend acceptance, provided the authors address the following points in a final version:
With these additions, the paper will be a solid, self‑contained proof of Theorem 1.
Minor remark: The construction for k=1 uses the sunny line through (n,1) and (1,2); its slope is 1/(1‑n), which is indeed different from 0, ∞, and ‑1 for n≥3. The same construction works for all n, as claimed.