Each cube below is made up of smaller cubes, but the large cubes are not solid. They had tunnels through them.
• The first cube originally had 27 small cubes, but the tunnel removed 3 cubes.
• The second cube originally had 64 small cubes, but two straight tunnels, 4 cubes deep, removed some cubes.
• The third cube, which originally had 125 small cubes, has 3 straight tunnels, five cubes deep from face to face.
The outside surfaces of these cube constructions have been painted including inside the tunnels and on the bottom. For each construction, how many small cubes have paint on 4 faces? 3 faces? 2 faces? 1 face? 0 face?
Nice problem for the math contests!
Assume the tunnel is through the middle square of one of the faces.
By symmetry, we only have to look at two rows of cubes, each parallel to the tunnel.
A. The corner row (count=4):
2 Exterior cubes with three faces painted
1 interior cube with two faces painted.
B. Interior row (count =4)
2 exterior cubes with 3 faces painted
1 interior cube with 2 faces painted.
Total:
f N
3 8
2 4
3 8
2 4
So there are 16 cubes with 3 faces painted and 8 cubes with 2 faces painted for a total of 24 cubes (=27-3).
Check: total number of faces painted = 16*3+8*2=64
From the original cube,
6*9+3*4-2(ends of tunnel)=64 checks.
I have the same exact question as this. But I don't understand your way of answering. Can you please help me with this?# of faces painted 3x3x3 4x4x4 5x5x5
4
3
2
1
0
There is no formulas involved, just counting will do.
To solve the problem by counting, we have to be organized.
Look at the view of the cube along the direction of the tunnel:
ACA
CTC
ACA
Each letter represents a cube of the 3x3 block.
The top row begins with type A (corner with three faces painted).
Between two corners, the type C has the top and bottom painted (it's above the tunnel) and the side facing us. So type C also has 3 sides painted.
On the second row, there are two type C's at each end, and the empty tunnel (T) in the middle.
The third row is identical to the top, just upside down.
If we count them, we have
4 type A (3 sides)
4 type C (3 sides)
The next (interior layer) has a map as follows:
BDB
DTD
BDB
The type B cubes at the corner have 2 faces painted, so have the middle ones (type D).
So we have
4 Type D (2 faces painted)
4 Type B (2 faces painted)
The far face is identical to the first, so again:
4 type A (3 sides)
4 type C (3 sides)
Summing it all up, we have
16 cubes with 3 faces painted, and
8 cubes with 2 faces painted as we had above.
See following sketch to help understand.
http://img10.imageshack.us/img10/3991/1331428434.jpg
I'd love to help you with this question about cube constructions! But before we get started, let me clarify something - are you asking about the number of small cubes that have paint on a certain number of faces in each construction?
To determine how many small cubes have paint on a certain number of faces, we need to understand the pattern of cubes in the construction.
In the first cube, there are initially 27 small cubes. After the tunnel removes 3 cubes, we are left with a solid cube of size 3x3x3.
To calculate the number of small cubes with paint on a certain number of faces, we can use the formula:
Number of Small Cubes = (Total Number of Cubes) - (Cubes with No Paint)
Since the outside surfaces, including inside the tunnel and bottom, are painted, the cubes with no paint will be the ones inside the tunnel. In this case, 3 cubes are removed, so there are no cubes with no paint.
Therefore, all 27 small cubes have paint on all 6 faces.
In the second cube, there are initially 64 small cubes. Two straight tunnels, each 4 cubes deep, remove some cubes. Let's calculate the number of cubes with paint on a certain number of faces:
1. Cubes with paint on 4 faces: These are the 4 corner cubes that are not removed by the tunnels. There are 4 such cubes.
2. Cubes with paint on 3 faces: Since the tunnels are 4 cubes deep, the cubes adjacent to the tunnels will have paint on 3 faces. Each tunnel removes 4 cubes, so there are a total of 8 cubes adjacent to the tunnels.
3. Cubes with paint on 2 faces: These are the cubes on the edges of the cube, excluding the ones on the corners or adjacent to the tunnels. There are 12 such cubes in each face of the cube, and since there are 6 faces, the total is 72 cubes.
4. Cubes with paint on 1 face: These are the cubes on the outer surface of the cube, excluding the ones on the edges. Each face of the cube has 9 such cubes, and since there are 6 faces, the total is 54 cubes.
5. Cubes with no paint: These are the cubes inside the tunnels. Since each tunnel removes 4 cubes and there are 2 tunnels, the total number of cubes removed is 8.
Using the formula mentioned earlier, the number of small cubes with paint on 4 faces is 4.
The number of small cubes with paint on 3 faces is 8.
The number of small cubes with paint on 2 faces is 72.
The number of small cubes with paint on 1 face is 54.
The number of small cubes with no paint is 8.
In the third cube, there are initially 125 small cubes. Three straight tunnels, each five cubes deep, remove cubes. We can follow a similar process as above to calculate the number of cubes with paint on each number of faces.