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 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 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 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 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.