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Packing a 3-dimensional object with 3-dimensional L-shaped pieces Question
Consider the following L-shaped 3-dimensional object consisting of 3 unit cubes joined at their faces:
Is it possible to use six of the above L-shaped pieces to form the following 3-dimensional object (whose volume is 18 unit cubes)?
If yes, show how the object can be formed.
If no, explain why not.
1 answer
Answer! Click or tap here to reveal
In terms of volume, six of the L-shaped pieces have the correct total volume to form the desired 3-dimensional object (whose volume is 18 unit cubes) but it is nevertheless impossible to form the desired object with six L-shaped pieces.
Justification! Click or tap here to reveal
Proof of answer:
Color seven of the unit cubes of the desired 3-dimensional object red as shown below:
Note: One of the red unit cubes at the bottom level of the desired object is not visible in the above diagram but its position can be determined in the following cross sectional view:
We only have six L-shaped pieces but there are seven unit cubes colored red. By the Pigeonhole principle, at least one of the L-shaped pieces must fill at least two red unit cubes. But the red unit cubes are too far apart to allow this.
Therefore it is not possible to use six of the L-shaped pieces to form the desired 3-dimensional object.
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