Question 1091573: An empty box with top, bottom and four sides, is a two and a half-foot cube. It is completely filled with smaller boxes that each measure ten inches on each side.
How many small boxes will be on the top layer in the big box?
How many small boxes will be touching the four sides of the big box?
How many small boxes are not against the top, bottom, or any side of the big box?
How many small boxes are on the bottom and also touching the sides of the big box?
How many small boxes are on the top and also touching two sides of the big box?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The big empty box is shaped like a cube with edges'lengths measuring
 
The easy way to calculate answers is to visualize how the box looks,
and how it could be filled.
You would start by covering the bottom of the big box with one layer of small boxes.
After that you would add more layers, one by one, until the box is full.
For each layer, you would start by putting
as many small boxes in a row as would fit against one of the cube's edges.
The number of boxes in a row would be
 small boxes.
How many small boxes will be on the TOP LAYER in the big box?
With  boxes against each edge,
each layer would have  small boxes,
so there would be  small boxes on the top layer
(and on each and every layer).
There would also be layers filling the big box, for a total of
 small boxes.
How many small boxes will be TOUCHING THE FOUR SIDES of the big box?
Each layer would be a 3 by 3 square small box arrangement,
with  small box at the center,
surrounded by the remaining small boxes around the layers' edges.
in each layer, that small box at the center would be the only one
not touching any sides.
The remaining  small boxes in each layer,
for a total of  small boxes
will be touching at least one side of the big box.
None could be touching all four sides, though.
How many small boxes are NOT AGAINST THE TOP, BOTTOM OR ANY SIDE of the big box?
We have found that there would be layers of small boxes,
each one consisting of  small boxes,
with one of them as the top layer.
Of course, another one of the layers would be the bottom layer,
so every one of the small boxes in those two layers
would be against the top or bottom of the big box.
The middle layer, would have small boxes that would not be
against the top or bottom of the big box.
However, the small boxes on each edge of that 3 by 3 square small box arrangement
would be against a side of the big box,
and that accounts for  of those small boxes.
Only the other  small box at the center of the middle layer
would be not against the top, bottom, or any side of the big box.
How many small boxes are ON THE BOTTOM AND ALSO TOUCHING THE SIDES of the big box?
As stated above,
exactly  small boxes on each layer (including the bottom layer)
are touching the sides of the big box.
How many small boxes are ON THE TOP AND ALSO TOUCHING TWO SIDES of the big box?
As stated above, each layer would be a 3 by 3 square small box arrangement,
with small box at the center,
surrounded by small boxes around the layers' edges.
Of the small boxes in each layer (including the top layer),
only the  small boxes on the corners
are touching TWO sides of the big box.
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