FIRST
Imagine a 4 by 4 by 4 cube hanging in front of you with just the
front face facing you. The cube is made up of 4 x 4 x 4, 64 cubes.
You drill holes through the four corner cubes, that are facing you,
all the way through to the back.
A friend looks down on the cube, from above, and they also drill
four holes through their four corner cubes all the way through to
the bottom.
You and your friend examine all the 64 small cubes.
You need to find out how many small cubes have holes in them and
how many have no holes in them.
Please justify your findings to someone else!
Then look at how many small cubes have two holes and how many just
one hole?
SECOND
Instead of drilling in the corners, a 2 x 2 square in the middle of
the face is chosen and these are drilled through to the back. The
person looking from above also chooses the middle part of the face
and drills through these to the bottom.
You and your friend examine all the 64 small cubes. You need to
find out how many small cubes have holes in them and how many have
no holes in them.
Please justify your findings to someone else!
Then look at how many small cubes have two holes and how many just
one hole?
THIRD
Now suppose the same things happen but this time the cube is a 5 x
5 x 5 cube:
The first part will be similar but for the second part you chose to
drill through the square that's just inside the edge, and drill
through all the small cubes that make up that square. (See
Hint if this is not clear)
Again the challenge is to work out how many small cubes have holes
and how many do not have holes.
Then look at how many small cubes have two holes and how many have
just one hole?
FOURTH
Go further with larger and larger cubes and examine relationships
and patterns, both
- between the two different kinds
of drilling
- and between different sizes of cubes.
FIFTH
Why stick to corner drilling and central square
drilling?
Consider large cubes made up of black and yellow cubes in a
chess board type pattern.
Then explore as before.