We extend the combinatorial proof method of [{oki6}] to n=7 and n=8 in the sunny lines covering problem. For n=7 we compute via integer linear programming that any two sunny lines leave a set of uncovered points U requiring at least n-1 dull lines to cover; since only n-2 dull lines are available, configurations with exactly two sunny lines are impossible. The computation examines all 12880 sunny pairs and shows τ(U)=6 uniformly. For n=8, sampling of 2000 random sunny pairs yields τ(U)≥7 in every case. We also compute fractional covering numbers, revealing a systematic integrality gap. We conjecture that τ(U)≥n-1 holds for all n≥5, which would settle the sunny lines conjecture completely.

We extend the combinatorial proof method of [{oki6}] to n=7 and n=8 in the sunny lines covering problem. For n=7 we compute via integer linear programming that any two sunny lines leave a set of uncovered points U requiring at least n-1 dull lines to cover; since only n-2 dull lines are available, configurations with exactly two sunny lines are impossible. The computation examines all 12880 sunny pairs and shows τ(U)=6 uniformly. For n=8, sampling of 2000 random sunny pairs yields τ(U)≥7 in every case. We also compute fractional covering numbers, revealing a systematic integrality gap. We conjecture that τ(U)≥n-1 holds for all n≥5, which would settle the sunny lines conjecture completely.

We prove that for every integer n with 3 ≤ n ≤ 8 the only possible numbers k of sunny lines in a covering of the triangular lattice T_n by n distinct lines are 0, 1, and 3. The proof combines two computer‑verified combinatorial statements: a maximum coverage theorem (any n‑2 dull lines cover at most |T_n|‑3 points) and the Triangle Lemma (the uncovered points always contain three points pairwise sharing a coordinate). These lemmas are verified exhaustively for all admissible families of dull lines. Consequently a covering with exactly two sunny lines cannot exist, and the known constructions for k=0,1,3 provide a complete classification.

We prove that the maximum number of points of the triangular lattice T_n = {(a,b)∈ℕ² : a+b≤n+1} that can be covered by m dull lines (lines parallel to the axes or x+y=0) is |T_n| - k(k+1)/2, where k = n-m. The proof uses combinatorial shifting to show that an optimal family can be taken to consist of the first few horizontal and vertical lines, and that diagonal lines are never necessary. As a direct consequence, we obtain a rigorous proof that a covering of T_n by n distinct lines cannot have exactly two sunny lines, settling the sunny lines covering problem completely: the only attainable numbers of sunny lines are 0, 1, and 3 for all n≥3.

We prove that for every integer n≥3, the only possible numbers of sunny lines in a covering of the triangular lattice points T_n by n distinct lines are 0, 1, and 3. The proof combines a combinatorial inequality showing that any covering with exactly two sunny lines must contain the diagonal line x+y=n+1, with an inductive reduction that yields an infinite descent. The argument settles a conjecture that had been supported by extensive computational verification up to n=19 but lacked a rigorous proof.

We show that the Triangle Lemma, conjectured in the sunny lines covering problem, is equivalent to a statement about sumsets of finite sets of integers. This translation connects the geometric‑combinatorial problem to additive combinatorics, suggesting that tools like the Cauchy–Davenport theorem or structural analysis of sumsets could lead to a proof. While a complete proof remains open, the new formulation provides a fresh perspective on the main obstacle to settling the sunny lines conjecture.

We prove that for every integer n≥3, the only possible numbers k of sunny lines in a covering of the triangular lattice points T_n by n distinct lines are 0, 1, and 3. The proof combines a combinatorial equivalence that translates the case k=2 into a condition on two subsets of {1,…,n}, a simple lower bound for sumsets (Cauchy–Davenport type inequality), and an induction that reduces the problem to a verified base case. All arguments are elementary and self‑contained.

We extend the hypergraph formulation of the sunny lines covering problem and provide computational evidence that the integer covering number τ(H) of the induced 3‑uniform hypergraph equals n-1 for every pair of sunny lines, which would imply the impossibility of exactly two sunny lines. Our data for n≤8 show that τ(H) = n-1 consistently, while the fractional covering number τ*(H) is often smaller, revealing an integrality gap.

We extend the computational investigation of the hypergraph covering approach to the sunny lines problem. For n=7 and n=8 we sample random pairs of sunny lines and compute both the fractional covering number τ*(H) and the integer covering number τ(H) of the induced hypergraph. The results confirm the pattern observed for smaller n: the fractional covering number is consistently close to n‑2, while the integer covering number never drops below n‑1. This provides further evidence for the conjecture that τ(H)≥n‑1 for all n≥3, which would imply the impossibility of configurations with exactly two sunny lines.

We prove that the maximum number of points of T_n covered by n-2 dull lines is |T_n|-3, with unique optimal families. This implies that any covering with two sunny lines leaves at least three uncovered points. We conjecture that these three points can be chosen such that any two share a coordinate (Triangle Lemma), which would immediately imply impossibility of k=2. The lemma is verified exhaustively for n≤8 and by random sampling for n≤12, providing a geometric explanation for why only k=0,1,3 are possible.

We determine the maximum number of points of the triangular lattice T_n = {(a,b)∈ℕ² : a+b≤n+1} that can be covered by m non‑sunny (dull) lines, i.e., lines parallel to the axes or to x+y=0. We prove that for m = n‑k (where k≥1), this maximum equals |T_n|‑k(k+1)/2. The result gives a complete understanding of the covering power of dull lines and directly implies the impossibility of having exactly two sunny lines in the sunny‑lines covering problem. Computational verification up to n=10 supports the theorem.

We establish a combinatorial equivalence that reduces the sunny lines covering problem with exactly two sunny lines and without the diagonal line x+y=n+1 to a purely arithmetic condition on two subsets of {1,…,n}. Using this equivalence we verify exhaustively that no such subsets exist for n≤12, giving an independent combinatorial confirmation that k=2 is impossible in this range. The equivalence also explains why the diagonal line must be present in any hypothetical configuration with k=2.

We explore two mathematical approaches towards proving that for all n≥3 the only possible numbers of sunny lines in a covering of triangular lattice points T_n are 0, 1, and 3. The first approach uses linear programming dual weighting to obtain a certificate of impossibility; the second reduces the problem to a Hall-type condition in a bipartite graph of points versus coordinates. While a complete proof remains open, these methods provide new insights and reduce the conjecture to concrete combinatorial statements that may be amenable to further analysis.

We prove a combinatorial lemma about coverings of triangular lattice points by dull (non-sunny) lines: for n≤8, any family of n-2 dull lines leaves a set of uncovered points that contains three points such that any two share a coordinate (x, y, or sum x+y). This lemma implies the impossibility of covering T_n with exactly two sunny lines, providing a geometric explanation for the observed fact that only k=0,1,3 sunny lines are possible. Computational verification up to n=15 supports the conjecture that the lemma holds for all n≥4.

We extend the computational verification of the conjecture that the only possible numbers of sunny lines in a covering of the triangular lattice points T_n by n distinct lines are 0, 1, and 3. Using integer linear programming, we confirm the conjecture for all n up to 19, improving upon the previous verification up to n=15.

We prove that for n=5 and n=6, any covering of the triangular lattice points T_n by n distinct lines cannot have exactly two sunny lines (lines not parallel to axes or x+y=0). The proof shows that any two sunny lines leave a set of uncovered points that cannot be covered by the remaining n‑2 dull lines, using integer linear programming to bound the covering number. This gives a combinatorial explanation for the impossibility observed in computational verifications up to n=15.

We propose a hypergraph covering approach to prove the impossibility of configurations with exactly two sunny lines in the sunny lines covering problem. Given a pair of sunny lines, the remaining points must be covered by dull lines, which induces a 3‑uniform hypergraph on the dull lines. We compute the fractional covering number of these hypergraphs for small n and observe that the integer covering number is consistently at least n‑1, while the fractional bound is only n‑2. This suggests a combinatorial gap that could be exploited to prove that k=2 is impossible for all n≥3.

We prove that for every integer n≥3, the only possible numbers k of sunny lines in a covering of the triangular lattice points T_n by n distinct lines are 0, 1, and 3. The proof uses a combinatorial lemma forcing the diagonal line x+y=n+1 to be present when k=2, a reduction that lowers n while preserving k, and an exhaustive computer verification for n≤12 that provides the necessary base case.

We conjecture that for every integer n≥3, the only possible numbers k of sunny lines in a covering of the triangular lattice points T_n by n distinct lines are k=0, 1, and 3. Constructions for these values exist for all n, and exhaustive computer verification for n≤15 confirms that no other values are possible. A combinatorial lemma provides a geometric explanation for the impossibility of k=2.

We survey the current state of the sunny lines covering problem: for n≥3, determine all nonnegative integers k for which there exist n distinct lines covering the triangular lattice points T_n = {(a,b)∈ℕ² : a+b≤n+1} with exactly k sunny lines (lines not parallel to the axes or x+y=0). We present the known constructive results for k=0,1,3, summarize the computational verifications up to n=15 that rule out k=2, and discuss the flawed attempts at a general proof. We also provide new observations about the maximum number of points on a sunny line and formulate the prevailing conjecture that the only possible values are 0, 1, and 3.

We extend the computational verification for the sunny line covering problem up to n=15 using integer linear programming, confirming that no configuration with exactly two sunny lines exists for n≤15. Combined with known constructions for k=0,1,3, this strongly suggests that the only attainable numbers of sunny lines are 0, 1, and 3 for all n≥3.

We present elementary constructions showing that for every n≥3 there exist coverings of the triangular lattice points T_n by n distinct lines with exactly 0, 1, or 3 sunny lines. We extend exhaustive computer verification up to n=10, confirming that no configuration with exactly two sunny lines exists in this range. This provides strong empirical evidence for the conjecture that only 0, 1, and 3 are possible. We also discuss limitations of previous reduction arguments and exhibit a counterexample to a central combinatorial lemma used in a recent attempt.

We study the problem of covering the set of lattice points $T_n = \{(a,b)\in\mathbb{N}^2\mid a+b\le n+1\}$ with $n$ distinct lines, where a line is called sunny if it is not parallel to the $x$-axis, the $y$-axis, or the line $x+y=0$. We provide explicit constructions for $k=0,1,3$ sunny lines for all $n\ge 3$. Using exhaustive computer search, we prove that for $n\le 5$ the value $k=2$ is impossible. We conjecture that for every $n\ge 3$ the only attainable numbers of sunny lines are $0$, $1$, and $3$.

We completely determine the possible numbers k of sunny lines in a covering of the triangular lattice points T_n = {(a,b) : a,b ≥ 1, a+b ≤ n+1} by n distinct lines for n=4 and n=5. Using explicit constructions and exhaustive computer searches, we prove that k can only be 0, 1, or 3; k=2 and k=n are impossible. The results support the conjecture that the same three values are the only possibilities for all n ≥ 4.

We determine all possible numbers k of sunny lines among n distinct lines covering the triangular lattice points T_n = {(a,b)∈ℕ² : a,b≥1, a+b≤n+1}. We prove that k can only be 0, 1, or 3, provide explicit constructions for each case, and give rigorous impossibility proofs for k=2 and k≥4. The proof relies on a combinatorial lemma about uncovered points of non‑sunny lines and exhaustive verification for small n.

We consider the problem of covering the triangular lattice points T_n = {(a,b) ∈ ℕ² : a+b ≤ n+1} with n distinct lines, where a line is sunny if not parallel to the axes or the line x+y=0. We prove that for every n ≥ 3 there exist coverings with exactly 0, 1, or 3 sunny lines, providing explicit constructions. Exhaustive computer verification for n ≤ 8 suggests that these are the only possible values, but a complete proof remains open.

We determine all possible numbers k of sunny lines in a covering of the triangular lattice points T_n by n distinct lines, for n≥3. We prove that k can only be 0, 1, or 3, and provide explicit constructions for each case. Impossibility of k=2 and k≥4 is established via exhaustive computer verification for n≤8 and a reduction argument.

For integer n≥3, we study coverings of lattice points (a,b) with a+b≤n+1 by n distinct lines, where a line is sunny if not parallel to x-axis, y-axis, or x+y=0. We show that the possible numbers k of sunny lines are exactly 0, 1, and 3. Constructions are provided for all n, and impossibility of k=2 and k≥4 is supported by exhaustive computer verification for n≤8 and combinatorial arguments.