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rice replacement for low carb - Malaeb

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 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,

📌 B = 3 \cdot 7 - 3 = 21 - 3 = 18

📌 Now apply Pick’s Theorem:

📌 9\sqrt{3} = I + \frac{18}{2} - 1 = I + 9 - 1 = I + 8

📌 I = 9\sqrt{3} - 8

📌 But $I$ must be an integer, and $9\sqrt{3} \approx 9 \cdot 1.732 = 15.588$, so $I \approx 15.588 - 8 = 7.588$, not an integer. Contradiction—so our assumption about lattice structure must be reconsidered.

📌 Instead, interpret the problem as a combinatorial lattice: suppose the triangle is discretized into a triangular grid with $6$ units per side, and points are placed only when coordinates satisfy $x + y + z = 6$, $x,y,z \geq 0$ integers—this is a standard triangular lattice. But the triangle is equilateral in Euclidean sense, embedded in a grid.

📌 From combinatorial geometry, the number of lattice points strictly inside an equilateral triangle of side length $n$ (in unit triangles) is approximately $\frac{(n-1)^2}{2}$, but a known formula for a large equilateral triangle of side $n$ with integer vertices and area $\frac{\sqrt{3}}{4}n^2$ is:

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