## Wooden Blocks

A solid, wooden, rectangular prism is constructed by gluing a number of one-inch wooden cubes face to face. When the wooden prism is viewed (after construction) so that only three of its faces are visible, exactly $231$ of the one-inch cubes cannot be seen. What are the TWO greatest possible number of wooden cubes used to construct the rectangular prism? You must have both values correct. By the way, the least value is $384$.
Source: mathcontest.olemiss.edu 3/25/2013

SOLUTION
Suppose we have $12$ cubes shown in the figure below arranged in the dimensions $x\times y\times z=3\times 4\times 1=12$ that we want to cover with other cubes so that the they are hidden when viewed from this angle.

We place $12$ cubes on the top to cover the top face, $3$ cubes to cover the left face and $4$ cubes to cover the right face as illustrated in the figure below

To fill out the rectangular prism we need to add $3$ cubes to the left face, $4$ cubes to the right face and $2$ cubes to the corner.
Number of cubes needed to cover the $12$ cubes
$12+2\left (3\right )+2\left (4\right )+2=28$
Total number of cubes used to build the rectangular prism
$28+12=40$
If we arranged the $12$ cubes in a different configuration $2\times 3\times 2=12$, we end up with a different covering as shown below

Number of cubes needed to cover the $12$ cubes
$6+3\left (2\right )+3\left (3\right )+3=24$
Total number of cubes used to build the prism
$24+12=36$
We observe that the flatter the cubes we want to hide are laid out the more cubes will be used to build the prism.

We are now ready to tackle the problem of hiding $3\times 7\times 11=231$ cubes. How many cubes are used to build the prism in this configuration?
Nummber of cubes needed to cover the $231$ cubes
$21+12\left (3\right )+12\left (7\right )+12=153$
Total number of cubes used to build the prism
$153+231=384$

Let’s flatten the $231$ cubes so that we can get the greatest possible number of cubes.
Configuration $21\times 11\times 1=231$
Number of cubes needed to cover $231$ cubes
$231+2\left (21\right )+2\left (11\right )+2=297$
Total number of cubes used to build the prism
$297+231=528$

Configuration $33\times 7\times 1=231$
Number of cubes needed to cover $231$ cubes
$231+2\left (33\right )+2\left (7\right )+2=313$
Total number of cubes used to build the prism
$313+231=544$

Configuration $77\times 3\times 1=231$
Number of cubes needed to cover $231$ cubes
$231+2\left (77\right )+2\left (3\right )+2=393$
Total number of cubes used to build the prism
$393+231=624$

Configuration $1\times 231\times 1=231$
Number of cubes needed to cover $231$ cubes
$231+2\left (1\right )+2\left (231\right )+2=697$
Total number of cubes used to build the prism
$697+231=928$
The two greatest possible numbers are $928$ and $624$.

Answer: $928$ and $624$.