A Workshop on Culmination of Blocks

1. Precocity- A Workshop on Culmination of Blocks

3. Introduction
• Faces = 6
• Edges = 12 (Two faces
meet at each edge)
• Corner/Vertex = 8
(Three faces meet at
each vertex)

4. Cutting of cubes
1 Cut = 2 Parts 2 Cuts on same
axis = 3 Parts
So, Generally we can say N parallel cuts or
N cuts on same axis leads to (N+1) parts

5. Cutting of cubes
• 1 Cut on X-axis leads to 2
parts on X-axis
• Similarly, 1 cut on Y-axis
leads to 2 parts on Y-axis
• Similarly, 1 cut on Z-axis
leads to 2 parts on Z-axis
• So if we combining these
cuts
Total parts = 2 X 2 X 2 = 8 parts on cube

6. Cutting of cubes
• So in general, we can say-
• If there are p, q and r cuts on x, y and z axis
respectively then total number of parts
= (p + 1)(q + 1)(r + 1) parts on cube

7. Examples
• If there are 5 cuts on each axis of cube, then it
leads to how many cubes?
• Cubes = (5 + 1)(5 + 1) (5 + 1) = 216
• A cube is cut into 343 small cubes. How many
numbers of cuts will be required?
• 343 = 7 X 7 X 7
• = (6 + 1) (6 + 1) (6 + 1)
• = 18 cuts

8. Examples
• A cube of edge 10 cm is cut into small cubes of
edge 2 cm.
(a) How many cuts will be required?
(b)How many small cubes will be obtained?
• (a) 10/2 = 5 X 5 X 5 = (4 + 1) (4 + 1) (4 + 1)
• = 12 cuts
• (b) Cubes = 5 X 5 X 5 = 125
• OR Cubes = (10/2)3 = 125

9. Examples
• A cuboid of dimension 8 X 10 X 12 Cm3 is cut into
small cubes of edges 2 Cm.
(a) How many cuts will be required?
(b) How many small cubes will be obtained?
• (a) Cuboid = 8 X 10 X 12
• (3+1)(4+1)(5+1)
• Cuts = 3 / 4 / 5 = 12
• (b) Cubes = 4 X 5 X 6 = 120

10. Example
• Consider 2 cubes painted blue on
all there faces. One of the cube is
cut into 27 small cubes and the
other is cut into 64 small cubes
then how many cubes will have-
(a) 3 faces painted (b) 2 faces
painted (c) 1 face painted (d) No
face painted
27 Cubes 64 cubes General
3 X 3 X 3 4 X 4 X 4 m X m X m
3 faces painted 8 8 8
2 faces painted 12 = 12 X (3-2) 24 = 12 X (4-2) 12 X (m-2)
1 faces painted 6 = 6 X (3-2) 2 24 = 6 X (4-2) 2 6 X (m-2)2
No face painted 1 23 (m-2)3

11. • A solid cube of each side 25 cm, has been painted red,
yellow and green on pairs of opposite faces, It is then
cut into cubical blocks of each side 5 cm. Answer the
following questions.
(1) Total number of cubes.
(2) How many cubes have three faces painted?
(3) How many cubes have only two faces painted?
(4) How many cubes have only one face painted?
(5) How many cubes have zero face painted?
(6) How many cubes have both red and green color?
(7) How many cubes both red and green color and not
yellow?
(8) How many cubes have only green color?
(9) How many cubes have two red painted face?

12. (1) Total number of cubes
=(25/5) 3
= 125 Cubes

13. (2) How many cubes have three
faces painted?
Ans. 8
(3) How many cubes have only
two faces painted?
Ans. = 12 X (m-2)
Here, m = 5,
Therefore, = 12 X (5-2) = 36

14. (4) How many cubes have only
one face painted?
Ans. = 6 X (m-2) 2
Here, m = 5,
Therefore, = 6 X (5-2) 2 = 54
(5) How many cubes have zero
face painted?
Ans. = (m-2) 3
Here, m = 5,
Therefore, = (5-2) 3 = 27

15. (6) How many cubes have both
red and green color?
• Ans. 20
(7) How many cubes both red
and green color and not
yellow?
• Ans. 12

16. (8) How many cubes have only
green color?
Ans. 18
(9) How many cubes have two
red painted face?
Ans. 0

17. Example
• What is the maximum possible number of
identical pieces a cube can be cut into by 7 cuts?
(a) 27 (b) 36 (c) 40 (d) 48
• The number of pieces obtained would be the
maximum possible when the given numbers of
cuts are made as equally as possible in three
directions.
• Distribution = 3, 2, 2
• No. of pieces = 4 x 3 x 3 = 36 (b)

18. Example
• What is the maximum possible number of
identical pieces a cube can be cut into by 9 cuts?
(a) 54 (b) 64 (c) 84 (d) 90
• The number of pieces obtained would be the
maximum possible when the given numbers of
cuts are made as equally as possible in three
directions.
• Distribution = 3, 3, 3
• No. of pieces = 4 x 4 x 4 = 64 (b)

19. Example
• How many cubes of dimensions 1 cm X 1 cm X
1 cm are required to cover a cube of
dimensions 6 cm X 6 cm X 6 cm completely?
(a) 127 (b) 296 (c) 386 (d) 216
• After covering the cube of dimensions 6 cm X
6 cm X 6 cm, the dimension of the cube will be
8 cm X 8 cm X 8 cm.
• So, (8 X 8 X 8) – (6 X 6 X 6) = 512 – 216 = 296(b).

20. Example
• How many cubes of dimensions 1 cm X 1 cm X
1 cm are required to cover a cuboid of
dimensions 6 cm X 7 cm X 8 cm completely?
(a) 343 (b) 296 (c) 384 (d) 420
• After covering the cube of dimensions 6 cm X
7 cm X 8 cm, the dimension of the cube will be
8 cm X 9 cm X 10 cm.
• So, (8 X 9 X 10) – (6 X 7 X 8) = 720 – 336 = 384(c).

21. Example
• How many cubes of dimensions 1 cm X 1 cm X 1
cm are required to cover a cuboid of dimensions
7 cm X 8 cm X 9 cm when it is placed at the
corner of room such that the three faces of the
cuboid are covered by two walls and the floor of
the room?
• (a) 169 (b) 216 (c) 256 (d) 420
• After covering the cube of dimensions 7 cm X 8
cm X 9 cm, the dimension of the cube will be 8
cm X 9 cm X 10 cm.
• So, (8 X 9 X 10) – (7 X 8 X 9) = 720 – 504 = 216(b).

22. Cutting of cubes along the two
diagonal
• If cubes are n X n X n.
• Two case are possible.
1. n is odd. 2. n is even.
If n is odd
n = 3 n = 5 n = odd
NUMBERS OF
CUT
9 + 9 -3 = 15 =
2 X 32 - 3
25 + 25 – 5 =
45 = 2 X 52 - 5
2n2 - n

23. Cutting of cubes along the two
diagonal
• If n is even
n = 4 n = EVEN
NUMBERS OF
CUT
4 X 4 + 4 X 4 =
2 X 42
2n2

24. Example
• 125 smaller cubes of dimensions 1 cm X 1 cm X 1
cm are stacked together to form a larger cube,
and then the cube is cut along two diagonals.
How many of the smaller cubes are cut into
smaller pieces?
• a) 10 b) 25 c) 45 d) 50
• 125 = 5 X 5 X 5, here n = 5, which is odd.
• When the cube is cut along two diagonals, the
number of pieces that get cut = 2n2 – n
• = 2 X 52 – 5 = 45(C).

25. Example
• 512 smaller cubes of dimensions 1 cm X 1 cm X 1
cm are stacked together to form a larger cube,
and then the cube is cut along one diagonal. How
many of the smaller cubes are cut into smaller
pieces?
• a) 64 b) 128 c) 90 d) 100
• 512 = 8 X 8 X 8, here n = 8, which is even.
• When the cube is cut along one diagonal, the
number of pieces that get cut = n2
• = 82 = 64(a).

26. Example
• What is the least possible number of cuts
required to cut a cube into 33 identical pieces?
a) 32 b) 11 c) 12 d) 31
• We get the least possible numbers of cuts when
the given number of pieces is factorized in such a
way that the factors are as equal as possible.
• 33 = 11 X 3 X 1
• The number of cuts required is = 10 + 2 + 0 =
12(c).

27. Example
• What is the least possible number of cuts
required to cut a cube into 160 identical pieces?
• a) 159 b) 13 c) 14 d) 24
• We get the least possible numbers of cuts when
the given number of pieces is factorized in such a
way that the factors are as equal as possible.
• 160 = 8 X 5 X 4
• The number of cuts required is = 7 + 4 + 3 = 14(c).

28. Example
• What is the least possible number of cuts
required to cut a cube into 72 identical pieces?
• a) 10 b) 9 c) 15 d) 11
• We get the least possible numbers of cuts when
the given number of pieces is factorized in such a
way that the factors are as equal as possible.
• 72 = 6 X 4 X 3
• The number of cuts required is = 5 + 3 + 2 = 10(a).

29. Example
• A cube is painted in such a way that the pair of adjacent faces
is painted in green; a pair of opposite faces is painted in
yellow and another pair of adjacent faces is painted in red.
This cube is now cut into 125 small but identical cubes. How
many small cubes have exactly two faces painted in green?
• a) 10 b) 7 c) 5 d) 8
• 5(c).

30. Example
• A cube is painted in such a way that the pair of adjacent faces
is painted in green; a pair of opposite faces is painted in
yellow and another pair of adjacent faces is painted in red.
This cube is now cut into 125 small but identical cubes. How
many small cubes have at least two different colors on their
faces?
• a) 30 b) 38 c) 36 d) 42
• 38(b).

31. Example
• A cube is painted in such a way that the pair of adjacent faces
is painted in green; a pair of opposite faces is painted in
yellow and another pair of adjacent faces is painted in red.
This cube is now cut into 125 small but identical cubes. How
many of the small cubes have exactly one color on them?
• a) 60 b) 45 c) 54 d) 15
• 60(a).

32. Example
• A cube is painted in such a way that the pair of adjacent faces
is painted in green; a pair of opposite faces is painted in
yellow and another pair of adjacent faces is painted in red.
This cube is now cut into 125 small but identical cubes. How
many of the small cubes do not have green color but have
yellow or red colors on them?
• a) 40 b) 75 c) 80 d) 53
• The number of small cubes with
yellow or red but not green =
total no. of small cubes – the
no. of small cubes with only
green – the no. of small cubes
with no color
• =125 – (25 + 20) – 27 = 53

33. Example
• A cube is painted in such a way that the pair of adjacent faces
is painted in green; a pair of opposite faces is painted in
yellow and another pair of adjacent faces is painted in red.
This cube is now cut into 125 small but identical cubes. How
many small cubes have exactly two painted faces and have
exactly two colors on them?
• a) 36 b) 30 c) 24 d) 34
• 30(b)

34. • The following questions are based on the information given below:
• A cuboid shaped wooden block has 4 cm length, 3 cm breadth and
5 cm height.
• Two sides measuring 5 cm x 4 cm are coloured in red.
• Two faces measuring 4 cm x 3 cm are coloured in blue.
• Two faces measuring 5 cm x 3 cm are coloured in green.
• Now the block is divided into small cubes of side 1 cm each.
• How many small cubes will have will have three faces coloured ?
• A.14 B.8 C.10 D.12
• Ans 8(B)

35. • The following questions are based on the information given below:
• A cuboid shaped wooden block has 4 cm length, 3 cm breadth and
5 cm height.
• Two sides measuring 5 cm x 4 cm are coloured in red.
• Two faces measuring 4 cm x 3 cm are coloured in blue.
• Two faces measuring 5 cm x 3 cm are coloured in green.
• Now the block is divided into small cubes of side 1 cm each.
• How many small cubes will have only one face coloured ?
• A.12 B.28 C.22 D.16
•
Ans:22
2 from the front + 2 from the back + 3
from the left + 3 from the right + 6 from
the top + 6 from the bottom = 22

36. • The following questions are based on the information given below:
• A cuboid shaped wooden block has 4 cm length, 3 cm breadth and
5 cm height.
• Two sides measuring 5 cm x 4 cm are coloured in red.
• Two faces measuring 4 cm x 3 cm are coloured in blue.
• Two faces measuring 5 cm x 3 cm are coloured in green.
• Now the block is divided into small cubes of side 1 cm each.
• How many small cubes will have no faces coloured ?
• A.None B.2 C.4 D.6
Ans:6
Required number of small cubes
= (5 - 2) x (4 - 2) x (3 - 2)= 3 x 2 x 1= 6

37. • The following questions are based on the information given below:
• A cuboid shaped wooden block has 4 cm length, 3 cm breadth and
5 cm height.
• Two sides measuring 5 cm x 4 cm are coloured in red.
• Two faces measuring 4 cm x 3 cm are coloured in blue.
• Two faces measuring 5 cm x 3 cm are coloured in green.
• Now the block is divided into small cubes of side 1 cm each.
• How many small cubes will have two faces coloured with red and
green colours ?
• A.12 B.8 C.16 D.20
Ans:12
Required number of small cubes
= 6 from the top and 6 from the bottom
= 12