3. A cube of side 4 cm is painted with 3 colours red, blue and green in such a way that
opposite sides are painted in the same colour. This cube is now cut into 64 cubes of equal
size. How many have at least two sides painted in different colours.
Answers
Answer:
30 cubes
Step-by-step explanation:
volume is 4^3=64 cc
so 64 cubes with side 1 cm
so at each side around the big cube, small cubes will be 12 as there are six sides so total 12x6=36 cubes
but 6 cubes are common so total cubes 30
Answer
The answer is 32 smaller cubes have at least two sides painted in different colours.
Explanation
- First, we need to find the number of smaller cubes that are obtained by cutting the big cube. Since the big cube has a side of 4 cm and the smaller cubes have a side of 1 cm, we can fit 4 smaller cubes along one edge of the big cube. So, the total number of smaller cubes is (4/1)^3 = 64.
- Next, we need to find how many smaller cubes have at least two sides painted in different colours. To do this, we can use a formula that relates the number of painted sides to the number of smaller cubes:
- No sided painted = (n-2)^3
- One sided painted = 6 * (n-2)^2
- Two sided painted = 12 * (n-2)
- Three sided painted = Always 8 (a cube has 8 corners)
3. In this formula, n is the number of smaller cubes along one edge of big cube. In our case, n = 4. So, we can plug in this value and get:
- No sided painted = (4-2)^3 = 8
- One sided painted = 6 * (4-2)^2 = 24
- Two sided painted = 12 * (4-2) = 24
- Three sided painted = Always 8
Therefore, the number of smaller cubes that have at least two sides painted in different colours is the sum of two sided and three sided painted cubes. That is:
- At least two sided painted = Two sided painted + Three sided painted
- At least two sided painted = 24 + 8
- At least two sided painted = 32
Hence, the answer is 32 smaller cubes have at least two sides painted in different colours.
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