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Difference between revisions of "2008 iTest Problems/Problem 88"

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

Latest revision as of 23:15, 18 July 2019

Problem

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?

Solution

<insert solution>

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