Cube Inscribed In Cube

Geometry Level 3

A cube is inscribed in a unit cube so that exactly 6 of the inscribed cube's 8 vertices each touch a different face of the unit cube.

What's the smallest cube that can be inscribed in this way?

If its side length can be expressed as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, give the product \(a \times b\) as the answer.

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