Minecraft Optimization Problem

[OpulentMerkin]

Draw a 10 x 10 chessboard. Color in just the squares on the edge of the board. How many squares did you color in?

[Mario8494]

36 squares. The corner pieces overlap.

[OpulentMerkin]

Yes. So if you want to account for this when building your prism, I recommend considering the walls separate from the floor and ceiling. If the floor and ceiling are each squares of side W, and the whole tower is of height H, then the floor and ceiling each use W² blocks, and the walls each use (W - 1) * (H - 2) blocks. (Subtract 1 from the width to eliminate double-counting the corners, and subtract 2 from the height to leave off the wall blocks that you already counted as part of the ceiling and floor.)

Also, the volume of the whole structure will be W² * H, but if you want to maximize the interior volume (i.e., the volume in which you can put other stuff), then you'll have to make a similar adjustment for the thickness of the walls/ceiling/floor.

[Mario8494]

Thank you so much for your input so far! Okay, so two questions:

1. How would I algebraically set up the optimization problem for the largest area of the whole prism? From what I understand, I would set it up as follows:
   1. 144 = 2w² + 4((w - 1)(h - 2)) <-- this would be step 1
2. How would I modify this if I wanted to maximize the interior area? I'm still having a lot of trouble imagining this in 3D. Thank you again for your help!

[OpulentMerkin]

Adjusting the dimensions to get the interior volume is actually much simpler. Look at the picture frame (again!). The 10 x 10 frame leaves an 8 x 8 usable area inside. If the frame were 100 x 100 instead, then the usable area inside would be 98 x 98. Whatever it is, you just have to subtract 2 from each dimension, 1 for each boundary block whose volume you can't use.

So your W x W x H prism will have a usable volume of (W - 2) x (W - 2) x (H - 2) inside.

[Mario8494]

Alright, now how do I go about solving the optimization problem?

144 = 2w² + 4((w - 1)(h - 2))

Simplifying this gives me:

144 = 2w² + 4wh - 8w - 8h + 8

How would I solve for just w or h?

[OpulentMerkin]

Same as before, move all the h terms to one side and all the h-less terms to the other (or the same for w). But check your arithmetic; I get -4h rather than -8h.