Reasoning: Concept of Cubes
Example:
A
cube of each side 4 cm, has been painted black, red and green on pairs of
opposite faces. It is then cut into small cubes of each side 1 cm.
The following questions and answers are based on the information give above:
1. How many small cubes will be there ?
Total
no. of cubes = (sides)3 = (4)3 = 64
2. How many small cubes will have three faces
painted?
From
the figure it is clear that small cubes having three faces coloured are
situated at the corners of the big cube because at these corners the three
faces of the big cube meet.
Therefore
the required number of such cubes is always 8, because there are 8 corners.
3. How many small cubes will have only two faces
painted ?
From
the figure, it is clear that to each edge of the big cube 4 small cubes are
connected and two out of them are situated at the corners of the big cube which
have all three faces painted.
Thus,
to edge two small cubes are left which have two faces painted. As the total no.
of edges in a cube are 12.
Hence
the no. of small cubes with two faces coloured = 12 x 2 = 24
(or)
No.
of small cubes with two faces coloured = (x - 2) x No. of edges
where
x = (side of big cube / side of
small cube)
Hence
in a generalized form, the total number of cubes with 2 sides painted = 12
(x-2)
4. How many small cubes will have only one face
painted ?
The
cubes which are painted on one face only are the cubes at the centre of each
face of the big cube.
Since
there are 6 faces in the big cube and each of the face of big cube there will
be four small cubes.
Hence,
in all there will be 6 x 4 = 24 such small cubes (or) (x - 2)2 x 6.
5. How many small cubes will have no faces painted?
No.
of small cubes will have no faces painted = No. of such small cubes
=
(x - 2)3 [Here x = (4/1) = 4 ]
=
(4 - 2)3
=
8.
6. How many small cubes will have only two faces
painted in black and green and all other faces unpainted ?
There
are 4 small cubes in layer II and 4 small cubes in layer III which have two
faces painted green and black.
Required
no. of such small cubes = 4 + 4 = 8.
7. How many small cubes will have only two faces
painted green and red ?
No.
of small cubes having two faces painted green and red = 4 + 4 = 8.
8. How many small cubes will have only two faces
painted black and red ?
No.
of small cubes having two faces painted black and red = 4 + 4 = 8.
In
the questions above, we had generalized the total number of cubes with exactly
2 colours as = 12 x (x-2).
Now,
there will be 3 such combinations of 2 colours ie black and red, green and red
and black and green. Therefore, the total number of cases will get divided into
3 equal parts.
Therefore,
for each combination, number of cases = (12/3) (x-2)
ie
4(x-2)
This
can be verified by looking at questions 6,7 and 8 above.
9. How many small cubes will have only black
painted ?
No.of
small cubes having only black paint. There will be 8 small cubes which have
only black paint. Four cubes will be form one side and 4 from the opposite
side.
10. How many small cubes will be only red painted ?
No.
of small cubes having only red paint = 4 + 4 = 8.
11. How many small cubes will be only green painted
?
No.
of small cubes having only green paint = 4 + 4 = 8.
In
the earlier questions, we have generalized the total number of cubes with one
side painted as = 6 (x-2)^2
There
will be 3 such cases ie only green, only black and only red. Therefore, each
case can be calculated as = (6/3) (x-2)^2
12. How many small cubes will have at least one
face painted ?
No.
of small cubes having at least one face painted = No. of small cubes having 1
face painted + 2 faces painted + 3 faces painted
=
24 + 24 + 8
=
56.
13. How many small cubes will have at least two
faces painted ?
No.
of small cubes having at least two faces painted = No. of small cubes having
two faces painted + 3 faces painted
=
24 + 8
=
32.
