A Very Big Block.

Number Theory Level 5

A bricklayer with too much time to spare has \(200,000\) bricks of size \(1 \times 2 \times 3\) units at his disposal.

He proposes to construct a solid cuboid block with these bricks (i.e., not just the walls the entire interior of the cuboid is filled with bricks). Further, no brick is broken for the construction and no part of any brick is to project out of the cuboid.

In order to make the problem interesting, he plans to construct the block so that in addition to the length, breadth and height, all the face diagonals also have integer lengths.

Will he be able to finish the construction with the bricks available with him?

Image credit:Wikipedia Steve Evans.

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