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Problem of the Day #62: An Infinite Structure of Cubes May 20, 2011

Posted by Billy in : potd , trackback

Albert has an infinite number of cubes, with side lengths $1$, $\frac{1}{2}$, $\frac{1}{4}$, etc. He aligns the largest cube such that each of its $6$ faces is parallel to either the $xy$, $yz$, or $xz$ plane. Albert constructs a structure in the following way: he picks up the next largest cube that he hasn’t used in his structure yet, chooses any face of the structure so far, and places the cube somewhere on that face. The structure begins with the cube of side length $1$, and Albert continues building until he has used up all the cubes. Let $x_{min}$ be the minimum $x$-coordinate of any of the vertices in his structure, and let $x_{max}$ be the maximum $x$-coordinate of any of the vertices in his structure. Define $y_{min}$, $y_{max}$, $z_{min}$, and $z_{max}$ similarly. What is the maximum possible value of $(x_{max}-x_{min})(y_{max}-y_{min})(z_{max}-z_{min})$?

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