Surface of a Cube is painted red and blue, 3 sides of each...

Question

Question: The surface of a cube is painted red and blue, 3 sides of each. The cube is cut into 27 equal sized smaller cubes. Find both answers to the question, "How many cubes have at least 2 sides painted blue?" and "How many cubes have at least 1 side painted red?" You must write down your solutions and make sure to turn in models that support those solutions.

Part of this problem is to use a physical model; drawings are required. I've already made a tangible 3-dimensional cube out of paper. 

 

For my answer, I got 5 total cubes that have at least 2 sides painted blue and 19 cubes that have at least 1 side painted red. Please help me with the answer, and how you got it.

 

Thank you!

Answer 1

Lets take the second question first as its fairly simple...

 

You have 3 sides, or faces of the larger cube painted red and three painted blue. Each face has 9 of the smaller cube faces showing, so that's 9 x 3 =27 cubes with at least one side painted red. This make sense when you figure that a cube has 6 faces, and each of the larger cube faces has 9 smaller faces in it. 6 times 9 = 54 total faces showing, and therefore PAINTABLE. 27 is half, which makes sense.

 

For the first question, if you look at a cube there are only two possible ways the sides can be configured: either the three sides painted blue are connected with one face connected to each of the other two along opposite edges(looking kind of like a letter C when you view them from the side), or the three sides are connected in one corner. In the first case you will have two edges of three cubes with two sides painted blue or 2x3= 6. In the second case you will have the corner piece with three faces painted blue, and each of the three edges connected to that corner having two more cubes with 2 sides painted blue. So in this case there you have 1 + 2*3 = 7 cubes with at least two sides painted blue. 

 

So the answer is 6 or7, and 27.