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Home » \(c_4 = 5 + 3 = 8\) - DNSFLEX

\(c_4 = 5 + 3 = 8\) - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Therefore, both systems will align at the start after $ \boxed{75} $ seconds.

📌 Question: A geographer is studying a triangular grid of weather stations where true equilateral triangles have sides aligned to a coordinate system based on modulo $6$ symmetry. If the distance between adjacent stations in one direction is $60$ km, how many integer-coordinate points (lattice points) lie strictly inside a large equilateral triangle with side length $6$ units (in km), assuming the vertices are at lattice points satisfying modular constraints?

📌 Solution: We model the triangular grid using a coordinate system adapted to equilateral triangles, but the problem simplifies: we interpret lattice points inside a large equilateral triangle of side length 6 in a triangular lattice where vertices are at lattice points and side length corresponds to 6 units. However, for computational geometry in such grids, the classic approach uses area and Pick’s Theorem.

📌 Pick’s Theorem states:

📌 A = I + \frac{B}{2} - 1

📌 where $A$ is area, $I$ is interior lattice points, and $B$ is boundary lattice points. We seek $I$, but the problem wants interior points only. First, compute the area of the large equilateral triangle with side length $6$:

📌 A = \frac{\sqrt{3}}{4} \cdot 6^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3}

📌 Now, determine $B$: the number of lattice points on the boundary. Each side has $6 + 1 = 7$ points, but corners are shared. Since it’s a triangle,

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