Since $x$ must be an integer, this is not possible. Re-evaluate using symmetry and integer constraints. Consider $(x, y, z) = \left(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\right)$ is known for symmetry, but integers require:

February 22, 2026

$Since $x$ must be an integer, this is not possible. Re-evaluate using symmetry and integer constraints. Consider $(x, y, z) = \left(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\right)$ is known for symmetry, but integers require:$

x^2 - 2x + 1 + y^2 + z^2 = 1 \quad \Rightarrow \quad x^2 + y^2 + z^2 - 2x + 1 = 1
Since $x^2 + y^2 + z^2 = 1$, we have:
- 2x + 1 = 1 \quad \Rightarrow \quad 2 - 2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}
Try $(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2})$ scaled to integers, $ (1, 1, 1) $ by symmetry and integer requirement.
Verify distances:
\sqrt{\left(1 - 0\right)^2 + \left(1 - 0\right)^2 + \left(1 - 0\right)^2} = \sqrt{3}

Since \(x\) must be an integer, this is not possible. Re-evaluate using symmetry and integer constraints. Consider \((x, y, z) = \left(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\right)\) is known for symmetry, but integers require:

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