Is it possible to build a $8×8×9$ block using $32$ bricks of dimensions $2×3×3$?

I tried to show that $8×8×9$ block can't contain $32$ blocks of dimensions $2×3×3$ . For that I tried to colour $1×1×1$ cubes.

(It would give me something like dominoes on chessboard where you can't use $1$ cell so blacks are more than whites , but domino covers the same number of blacks and whites)

I thought that I have to colour them in $18$ colours , but that's too much , and impossible to visualise in three-dimensional space.


The large block will have an $8\times 8$ face. It must break down into $2\times3$ and $3\times3$ faces. So each small face has area divisible by $3$, yet the total area of that face, $64$, is not divisible by $3$.

So it cannot be done.


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