CHAPTER 12: CUBES AND DICE:
NO OF CUBES: N3
ONE SIDE PAINTED (CENTRAL CUBE) = (N-2)2 × 6
TWO SIDE PAINTED (MIDDLE CUBE) = (N-2) × 12
NO SIDE PAINTED (INNER CUBE) = (N-2)3
THREE SIDE PAINTED (CORNER CUBE) ALWAYS 8
A CUBE HAS 6 FACES, 12 EDGES AND 8 CORNERS. WE CAN SEE THAT THE CUBES WHICH GOT ALL THE THREE SIDES PAINTED LIES AT THECORNER HENCE 8. CUBES WITH 2 SIDE PAINTED LIES ON THE EDGES BUT THOSE WHICH LIES ON THE LEFT AND RIGHT EDGE MATCHES WITH THE CORNER HENCE THESE 2 GETS SUBTACTED.
CUBES WITH ONE SIDE LIES ON THE SURFACE SINCE TOP AND BOTTOM ROW, LEFT AND RIGHT COLUMN MATCHES WITH THE EDGES HENCE THESE ARE EXCLUDED WHILE CALCULATING SINGLE SIDE PAINTED
A cube is a 3-dimensional diagram with all sides equal. If we divide it into the size (1⁄n)th part of its side, we get n3 smaller cubes.
Shown below is a cube which is painted on all the sides and the cut into (1⁄4)th of its original side.
Some observations: A cube has 6 faces, 12 edges and 8 corners. We can see that the cubes which got all the three sides painting lies at the corners. So the number of cubes which got painted all the three sides is equal to 8. Cubes with 2 sides painting lie on the edges (see the diagram). But the cubes which are on the left and right side of the edge matches with the corners. So we have to substract these two cubes from the number of cubes lying on the edge to get the number of cubes with 2 sides painting. Cubes with 1 side painting lies on the surfaces. Since, the top row, bottom row, left column, and right column matches with the edges, We must exclude these cubes while calculating the single side painted cubes.
The following rules may be helpful: If a cube is divided into the size (1n)th of its original side after get painted all the sides, Then
Total number of cubes = n3
Cubes with 3 sides painting = 8
Cubes with 2 sides painting = 12×(n−2)
Cubes with 1 sied painting = 6×(n−2)2
Cubes with no painting = (n−2)3
Solved Examples (Level - 1)
1. A cube whose two adjacent faces are coloured is cut into 64 identical small cubes. How many of these small cubes are not coloured at all?
Assume the top face of the cube and its right side are colored green and orange respectively.
Now If we remove the colored faces, we are left with a cuboid, whose front face is indicated with dots.
So on the front face there are 9 cubes, and behind it lies 4 stacks. So total 9 x 4 = 36
2. A cube, painted yellow on all-faces is cut into 27 small cubes of equal size. How many small cubes got no painting?
Assume we have taken out the front 9 cubes. Then the cube looks like the one below.
Now the cube which is in the middle has not got any painting. The cubes on the Top row, bottom row, left column and right column all got painting on atleast one face.
Alternative method:
Use formula: (n−2)3 Here n = 3 So (3−2)3 = 1
3. All surfaces of a cube are coloured. If a number of smaller cubes are taken out from it, each side 1/4 the size of the original cube's side, Find the number of cubes with only one side painted.
The original (coloured) cube is divided into 64 smaller cubes as shown in the figure. The four central cubes on each face of the larger cube, have only one side painted. Since, there are six faces, therefore total number of such cubes = 4 x 6 = 24.
Alternative Method:
Use formula : 6×(n−2)2 = 6×(4−2)2 = 24
Level - 2
4. Directions: One hundred and twenty-five cubes of the same size are arranged in the form of a cube on a table. Then a column of five cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are coloured red. Now, answer these questions based on the above statement:
(1) How many small cubes are there in the solid after the removal of the columns?
(2) How many cubes do not have any coloured face?
(3) How many cubes have only one red face each?
(4) How many cubes have two coloured faces each?
(5) How many cubes have more than 3 coloured faces each?
The following figure shows the arrangement of 125 cubes to form a single cube followed by the removal of 4 columns of five cubes each.
When the corner columns of the original cube are removed , and the resulting block is coloured on all the exposed faces (except the base) then we get the right hand side diagram. We labelled the various columns from a to u as shown in the figure
(1): Since out of 125 total number of cubes, we removed 4 columns of 5 cubes each, the remaining number of cubes = 125 - (4 x 5) = 125 - 20 = 105.
(2): Cubes with no painting lie in the middle. So cubes which are blow the cubes named as s, t, u, p, q, r, m, n, o got no painting. Since there are 4 rown below the top layer, total cubes with no painting are (9 x 4) = 36.
(3): There are 9 cubes namaed as m, n, o, p, q, r, s, t and u in layer 1, and 4 cubes (in columns b, e, h and k) in each of the layers 2, 3, 4 and 5 got one red face. Thus, there are 9 + (4 x 4) = 25 cuebs.
(4) the columns (a, c, d, f, g, i, j, l) each got 4 cubes in the layers 2, 3, 4, 5. Also in the layer 1, h, k, b, e cubes got 2 faces coloured. so total cubes are 32 + 4 = 36
(5): There is no cube in the block having more than three coloured faces. There are 8 cubes (in the columns a, c, d, f, g, i, j and l) in layer 1 which have 3 coloured faces. Thus, there are 8 such cubes.
Thus, there are 8 such cubes.
5. Directions: A cube of side 10 cm is coloured red with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 smaller cubes of equal size. Answer the following questions based on this statement:
(1) How many cubes have three green faces each?
(2) How many cubes have one face red and an adjacent face green?
(3) How many cubes have at least one face coloured?
(4) How many cubes have at least two green faces each?
Clearly, upon colouring the cube as stated and then cutting it into 125 smaller cubes of equal size we get a stack of cubes as shown in the following figure.
The figure can be analysed by assuming the stack to be composed of 5 horizontal layers.
(1): All the corner cubes are painted green. So there are 8 cubes with 3 sides painted green.
(2): There is no cube having one face red and an adjacent face green as all the green painted cubes got paint on atleast 2 faces.
(3): Let us calculate the number of cubes with no painting. By formula, (n−2)3 = (5−2)3 = 27
Therefore, there are 125 - 27 = 98 cubes having at least one face coloured.
(4): From the total cubes, Let us substract the cubes with red painting, cubes with no painting.
125 - (9 x 6) - 27 = 44
Construction of Boxes:
The details of the cube formed when a sheet is folded to form a box:
Form I
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In this case:
1 lies opposite 5;
2 lies opposite 4;
3 lies opposite 6.
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Form II
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In this case:
1 lies opposite 6;
2 lies opposite 4;
3 lies opposite 5.
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Form III
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In this case:
1 lies opposite 4;
2 lies opposite 6;
3 lies opposite 5.
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Form IV
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In this case:
1 lies opposite 4;
2 lies opposite 5;
3 lies opposite 6.
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Form V
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In this case:
1 lies opposite 3;
2 lies opposite 5;
4 lies opposite 6.
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Form VI
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In this case:
 will be the one of the faces of the cube and it lies opposite 3;
2 lies opposite 4;
1 lies opposite 5.
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Form VII
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In this case:
 will be the one of the faces of the cube and it lies opposite 3;
2 lies opposite 4;
1 lies opposite 5.
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Form VIII
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In this case:
 and  are two faces of the cube that lie opposite to each other.
1 lies opposite 3;
2 lies opposite 4;
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QUESTIONS ON CUBES:
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A cube whose two adjacent faces are colored is cut into 64 identical small cubes. How many of these small cubes are not colored at all?
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A cube, painted yellow on all faces is cut into 27 small cubes of equal size. How many small cubes got no painting?
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All surfaces of a cubes are colored. If a number of smaller cubes are taken out from it, each side ¼ the size of the original cube’s side, find the no of cubes with only 1 side painted. (after division 64 smaller cubes)
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One hundred and twenty five cubes of the same size are arranged in the form of a cube on a table. Then a column of 5 cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are colored red. Now, answer these questions based on the above statement:
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How many small cubes are there in the solid after the removal of the columns?
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How many cubes do not have any colored face?
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How many cubes have only one red face each?
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How many cubes have two colored faces each?
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How many cubes have more than three colored faces each?
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A cube of side 10 cm is colored red with a 2 cm wide green strip along the sides on all the faces. The cube is cut into 125 smaller cubes of equal size. Answer the following questions based on these statements:
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How many cubes have three green faces each?
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How many cubes have one face red and an adjacent face green?
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How many cubes have at least one face colored?
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How many cubes have at least 2 green faces each?
DICE:
S
T
F
FOR OUR UNDERSTANDING PURPOSE THE SIDE MARKED AS T WE WILL CAONSIDER AS TOP THE ONE WITH F AS FACE AND THE ONE WITH S AS SIDE.
SOME IMPORTANT POINTS:
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TWO OPPOSITE FACE CANNOT BE ADJACENT TO ONE ANOTHER.
4
4
3
5
2
6
WHICH NO WILL APPAER ON THE FACE OPSSOSITE TO THE FACE WITH NO 4. SOLUTION: SINCE 6, 2 5 3 ARE ADJACENT THEREFORE DEFINITELY 1 WILL BE OPPSOIT 4.
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IF TWO DIFFERENT POSITIONS OF A DICE ARE SHOWN AND ONE OF THE TWO COMMON FACE IS IN THE SAME POSITION, THEN THE REMAINING FACES WILL BE OPPOSITE TO ECAH OTHER.
5
4
5
3
3
2
ILLUSTRATION: HERE IN THE TWO DICES SHOWN NO 5 AND 3 ARE COMMON IN BOTH THESE DICES. HENCE AS PER THIS RULE THE TWO REMAINIG NO’S I.E 2 AND 4 WILL BE OPPOSITE TO EACH OTHER.
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IF IN THE TWO DIFFERENT POSITIONS OF A DICE, THE POSITION OF A COMMON FCAE BE THE SAME, THEN EACH OF THE OPPOSITE FACES OF THE REMAING FACES WILL BE IN THE SAME POSITION.
6
2
3
4
5
3
ILLUSTRATION: HERE IN BOTH POSITION THE COMON 3 IS IN THE SAME POSITION (F), HENCE AS PER RULE 5 WILL BE OPPOSITE 6 AND 4 WILL BE OPPOSITE 2.
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IF IN TWO DIFFERENT POSITION OF A DICE, THE COMMON FACE IS NOT THE SAME, THEN THE OPPOSITE FACE OF THE COMMON FACE WILL BE THAT WHICH IS NOT SHOWN ON ANY FACE IN THESE TWO POSITIONS. THE OPPOSITE FACE OF THE REMAINING FACE WILL NOT BE SAME.
1
4
2
3
1
5
ILLUSTRATION: HERE IN TWO POSITION THE FACE COMMON IS NO 1 AND IT IS NOT IN THE SAME POSITION (T IN ONE AND S IN THE OTHER). THE FACE WITH NO 6 IS NOT A PART OF ANY DICE, HENCE AS PER RULE 1 WILL BE OPPOSITE 6. 3 WILL BE OPPOSITE 2 AND 4 WILL BE OPPOSITE 5 ( AS PER RULE PART OF DIFFERENT DICE OPPOSITE TO ECAH OTHER)
QUESTIONS:
1.
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Which symbol will be on the face opposite to the face with symbol * ?
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@ $ 8 --
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2.
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Two positions of dice are shown below. How many points will appear on the opposite to the face containing 5 points?
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3 1 2 4
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3.
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Which digit will appear on the face opposite to the face with number 4?
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3 5 6 EITHER 2 OR 3
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4.
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Two positions of a dice are shown below. Which number will appear on the face opposite to the face with the number 5?
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EITHER 2 OR 6, 2 6 4
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5.
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How many points will be on the face opposite to in face which contains 2 points?
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1 5 4 6
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6.
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Which number is on the face opposite to 6?
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4 1 2 3
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7.
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Two positions of a dice are shown below. When number '1' is on the top. What number will at the bottom?
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3 5 2 6
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8.
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Two positions of a cube with its surfaces numbered are shown below. When the surface 4 touch the bottom, what surface will be on the top?
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1 2 5 6
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9.
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Here two positions of dice are shown. If there are two dots in the bottom, then how many dots will be on the top?
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2 3 5 6
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10.
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Two positions of dice are shown below. How many points will be on the top when 2 points are at the bottom?
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6 5 4 1
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11.
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Here 4 positions of a cube are shown. Which sign will be opposite to '+' ?
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% -- $
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12.
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Two positions of a cubical block are shown. When 5 is at the top which number will be at bottom?
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1 2 3 4
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13.
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From the four positions of a dice given below, find the color which is opposite to yellow ?
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Violet RED ROSE BLUE
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14.
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When the digit 5 is on the bottom then which number will be on its upper surface?
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1 3 4 6
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15.
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How many points will be on the face opposite to the face which contains 3 points?
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2 5 4 6
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16.
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Observe the dots on the dice (one to six dots) in the following figures. How many dots are contained on the face opposite to the containing four dots?
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2 3 5 6
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17.
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Two positions of a dice are shown below. When 3 points are at the bottom, how many points will be at the top?
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2 5 4 6
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18.
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From the positions of a cube are shown below, Which letter will be on the face opposite to face with 'A'?
D B C F
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Six dice with upper faces erased are as shows.
The sum of the numbers of dots on the opposite face is 7.
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1.
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If even numbered dice have even number of dots on their top faces, then what would be the total number of dots on the top faces of their dice?
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12 14 18 24
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2.
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If the odd numbered dice have even number of dots on their top faces, then what would be the total number of dots on the top faces of their dice?
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8 10 12 14
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3.
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If dice (I), (II) and (III) have even number of dots on their bottom faces and the dice (IV), (V) and (VI) have odd number of dots on their top faces, then what would be the difference in the total number of top faces between there two sets?
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0 2 4 6
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4.
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If the even numbers of dice have odd number of dots on their top faces and odd numbered dice have even of dots on their bottom faces, then what would be the total number of dots on their top faces?
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12 14 16 18
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5.
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If the dice (I), (II) and (III) have even number of dots on their bottom faces, then what would be the total number of dots on their top faces?
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7 11 12 14
DIRECTIONS TO SOLVE:
The figure given on the left hand side in each of the following questions is folded to form a box. Choose from the alternatives (1), (2), (3) and (4) the boxes that is similar to the box formed.
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1.
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2 and 3 only
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1, 3 and 4 only
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2 and 4 only
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1 and 4 only
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2.
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1, 2 and 4 only
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3 and 4 only
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1 and 2 only
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1, 2 and 3 only
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3.
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1 only
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2 only
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3 only
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4 only
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4.
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1 and 3 only
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2 and 4 only
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2 and 3 only
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1, 2, 3 and 4
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5.
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1 and 2 only
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2 and 3 only
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1, 2, 3 and 4
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2 and 4 only
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6.
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1 and 2 only
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2 and 4 only
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2 and 3 only
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1 and 4 only
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6.
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1 and 2 only
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2 and 4 only
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2 and 3 only
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1 and 4 only
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In a cube or a cuboid there are six faces in each.
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In a cube length, breadth and height are same while in cuboid these are different.
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In a cube the number of unit cubes = (side)3.
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In cuboid the number of unit cube = (l x b x h).
Example:
A cube of each side 4 cm, has been painted black, red and green on pars 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 the small cube having three faces colored are situated at the corners of the big cube because at these corners only 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 colored = 12 x 2 = 24
(or)
No. of small cubes with two faces colored = (x - 2) x No. of edges
where x = (side of big cube / side of small cube)
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.
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.
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.
The following questions are based on the information given below:
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A cuboid shaped wooden block has 6 cm length, 4 cm breadth and 1 cm height.
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Two faces measuring 4 cm x 1 cm are colored in black.
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Two faces measuring 6 cm x 1 cm are colored in red.
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Two faces measuring 6 cm x 4 cm are colored in green.
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The block is divided into 6 equal cubes of side 1 cm (from 6 cm side), 4 equal cubes of side 1 cm (from 4 cm side).
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How many cubes having red, green and black colors on at least one side of the cube will be formed ?
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