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.

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , . Bookmark the permalink.

One Response to Wooden Blocks

  1. I would have never thought it possible but you have just made something so simple ( wooden blocks ) seem so complicated (algebra). Your maths is great !

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