SOLUTION: How to find the number of cubes in an arrangement given the top, front and side views of an arrangement? Are there any strategies or formulas for finding it?

Question 1031009: How to find the number of cubes in an arrangement given the top, front and side views of an arrangement? Are there any strategies or formulas for finding it?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Assuming that each cube that is not at the bottom is directly on top of another cube,
the top view will let us know how the cubes on the bottom layer are arranged.
Each layer above that can be guessed from the front and side views.
However, with these top, front and side views,
,
how do we know which of the drawing below corresponds to the top view of the top layer?

FURTHER EXPLANATION ADDED AT A LATER DATE:
We can count 16 cubes on a bottom layer looking like this:
On the top view we see some (7 to 13) of those first layer cubes, AND
the cubes that are on top of the rest of the first layer.

The front and right side pictures tell us there are some cubes on a second layer.
.The front and side views show us 3 to 5 second layer cubes,
each labeled with one or two letters as
, , , , , and .
The problem is we do not know where those labeled cubes are within the second layer.
We do not know what cubes are hiding "behind" them, either.
All we know is that the second layer has at least cubes,
and at most cubes.
Cube could be on top of cube ,
or it could be on top of cubes , or .
Without any hint of perspective, we cannot tell.
If is on top of ,
are there other unseen cubes on top of cubes , and/or ?
We don't know.
Something like that happens with cubes and .
Regarding the cubes on the right side view,
Cube could be on top of cube ,
or it could be on top of cubes or .
Something like that happens with cubes and .
The second layer could have 9 cubes:
cube on top of cube ,
cube on top of cube ,
a cube on top of cube (labeled as on the front view and as on the right side view),
cube on top of cube ,
cube on top of cube ,
and another four cubes on top of cubes , , , and ,
that are not visible in the front or side views.
The top layer could have as few as 3 cubes,
strategically arranged so as to appear as cubes through on the front and side views.
One such arrangement of 3 second layer cubes could have
a cube on top of cube (labeled as and on the front and side views),
a cube on top of cube (labeled as and on the front and side views), and
a cube on top of cube (labeled as and on the front and side views).
Five other possibilities for a 3-cube second layer exist,
wit cubes on top of cubes
2, 8, and 1, or
3, 6, and 12, or
3, 8, and 10, or
4, 7, and 10, or
4, 6, and 11.
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