You have $1,000,000$ cubes of eight different colors, and you arrange them into a $2 \times 2 \times 250,000$ cuboid, so that no two cubes of the same color meet at a side, an edge or a vertex.

Is it possible for any two of the eight corner cubes of the $2 \times 2 \times 250,000$ cuboid to share the same color?

No
Yes

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With this arrangement, every row of $2 \times 2$ cubes needs to necessarily contain different colors than the row before it. So, if we call the colors on one end $1$ , $2$ , $3$ , and $4$ . Then every other row must contain the colors $5$ , $6$ , $7$ , and $8$ . Since there are an even number of rows, this means that $\boxed{\text{no}}$ there is no way that cube on one end can have the same color as a cube on the other end.