Question

A cube of 5 cm has been painted on its surfaces in such a way that two opposite surfaces have been painted blue and two adjacent surfaces have been painted red. Two remaining surfaces have been left unpainted. Now the cube is cut into smaller cubes of side 1 cm each.

a)How many cubes will have no side painted ?

b)How many cubes will have at least red colour on its surfaces ?

c)How many cubes will have at least blue colour on its surfaces ?

d)How many cubes will have only two surfaces painted with red and blue colour respectively ?

e)How many cubes will have three surfaces coloured?

Answer 1

The cube problem involves counting the number of smaller cubes with certain paint configurations after cutting. No side painted cubes are 27, cubes with at least red or blue colour are 26 each, two surface painted cubes are 1, and three surface coloured cubes are 4.

Step-by-step explanation:

We are dealing with a cube with a side of 5 cm that is cut into smaller 1 cm cubes. The original cube has two opposite surfaces painted blue, two adjacent surfaces painted red, and two surfaces unpainted. Our goal is to count the number of certain types of smaller cubes after the cut.

• No side painted: There will be 27 small cubes with no sides painted. These are from the innermost layer, as all the outer cubes will be painted on at least one side.
• At least red colour: There will be 9 + 9 + 3 + 3 + 1 + 1= 26 small cubes with at least one red side.
• At least blue colour: There will also be 26 cubes with at least one blue side (same calculation method as red).
• Only two surfaces painted red and blue: There will be 1 cube with exactly two surfaces painted, where one is red and one is blue.
• Three surfaces coloured: There will be 4 cubes at each of the four corners where the three painted surfaces meet, making a total of 4 cubes.