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Home » Solve for \( x \) in the equation \( 2^x+1 = 32 \). - IX Labs

Solve for \( x \) in the equation \( 2^x+1 = 32 \). - IX Labs

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 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,

📌 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.

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