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 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 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 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 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 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.