Question number 30. Basil used small cubes with a side length of 1 to construct a cube with a side length of 4. After that, he painted 3 faces of the big cube red and the other 3 faces blue. After he finished, there was no small cube with 3 red faces. How many small cubes have both red and blue faces? So here is a cube. Let's imagine that this is the large cube, constructed of lots and lots of smaller cubes. And I'll draw one smaller cube here in the corner to represent a possible coloring of the corner, like so. Now the way Basil colored this cube is that no face with red is joining faces with red. So let's begin by coloring the top face here in red. And if we do that, we're not allowed to color the adjoining faces, any of them in red. So both of them must be here colored in blue, like that. And now that will allow me to outline the general coloring scheme. I'll outline the blue faces in blue here. So the left face would be in blue. The front face, as we see in our little example with the small cube, is colored in blue. And also the right face would also have to be colored blue. The remaining 3 faces are then in red. Now we have here some neighboring faces that are both red and blue. That's where the cubes with both colors exist. And those edges where the faces meet up, like that, I'll outline in orange. So taking a few from our example here, the top edge of the front face is where red meets blue. Also by symmetry, the same thing happens at the bottom. The top edge of the right face has that property, by symmetry, also at the bottom. And the top edge of the left face is where red meets blue, likewise at the bottom. And here the back face, being red, meets up with the left and right faces, which are blue. So the edge I'll outline here in orange also contains mixed-color cubes. Okay, so what do we have? We have exactly 8 edges with cubes of both colors. So let's write that down. 8 edges have cubes of both colors. And so counting them, we could possibly overcount the corner cubes because they share faces. So we have the 8 corner cubes. And also an additional 2 cubes per edge. Those lie between the corners. So that's 8 plus 2 times 8 is 3 times 8, or 24 cubes have both colors. And so we have counted them all. And the answer here to number 30 would be that there are 24, or D, cubes with both red and blue faces.

cube

painting

red and blue faces

edge analysis

smaller cubes