2008 iTest Problems/Problem 88


A six dimensional "cube" (a $6$-cube) has $64$ vertices at the points $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3)$. This $6$-cube has $192\text{ 1-D}$ edges and $240\text{ 2-D}$ edges. This $6$-cube gets cut into $6^6=46656$ smaller congruent "unit" $6$-cubes that are kept together in the tightly packaged form of the original $6$-cube so that the $46656$ smaller $6$-cubes share $2-D$ square faces with neighbors ($\textit{one}$ $2$-D square face shared by $\textit{several}$ unit $6$-cube neighbors). How many $2$-D squares are faces of one or more of the unit $6$-cubes?


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