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Home » Thus, the sum becomes a telescoping series: - Silent Sales Machine

Thus, the sum becomes a telescoping series: - Silent Sales Machine

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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AB = \sqrt{(1-0)^2 + (0-1)^2 + (0-0)^2} = \sqrt{1 + 1} = \sqrt{2}
So all edges must be $\sqrt{2}$. Set up equations for $AD = BD = CD =...
$AD^2 = (x - 1)^2 + y^2 + z^2 = 2$
$BD^2 = x^2 + (y - 1)^2 + z^2 = 2$
$CD^2 = x^2 + y^2 + (z - 1)^2 = 2$
Subtracting equation (1) from (2):
[x^2 + (y - 1)^2 + z^2] - [(x - 1)^2 + y^2 + z^2] = 0
x^2 + y^2 - 2y + 1 + z^2 - (x^2 - 2x + 1 + y^2 + z^2) = 0
-2y + 2x = 0 \quad \Rightarrow \quad x = y
Similarly, subtract (1) from (3):

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