A rectangular cube with sides measuring 4 cm, is made with 1 cm^3 blocks. Two straight tunnels of 4 cubes each are taken out. All exposed surfaces are painted including inside the tunnels. How many small cubes have been painted on exactly 3 faces?
The original cube has 8 corners; each of them will be painted of 3 faces. The question then is how many other cubes will be painted on 3 faces after the two tunnels are taken out.
The problem says that both tunnels are 4 cubes, so that means the tunnels do not intersect.
However, there are different ways to take out two non-intersecting tunnels of 4 cubes each, and they produce different numbers of additional faces that will be painted on 3 sides.
So the statement of the problem is not sufficiently precise to answer the question. But your answer should be 19
Well, this is a tricky question! Let's dive into it, shall we?
First, let's visualize what we have. We have a rectangular cube with sides measuring 4 cm, made up of 1 cm^3 blocks. We have two tunnels consisting of 4 cubes each, and all exposed surfaces, including inside the tunnels, are painted.
Now, let's examine each face. There are six faces on a cube. For each of these faces, we need to determine how many small cubes have been painted on exactly 3 faces.
If we take a closer look at the tunnels, we can see that each tunnel has four cubes. Since all the exposed surfaces, including inside the tunnels, are painted, this means that all sides of the cube that face the tunnels have been painted. Therefore, each tunnel contributes 4 small cubes that have been painted on exactly 3 faces.
Since we have two tunnels, that means we have a total of 8 small cubes that have been painted on exactly 3 faces.
So, to answer your question, there are 8 small cubes that have been painted on exactly 3 faces.
To find the number of small cubes that have been painted on exactly 3 faces, we need to consider the arrangement of cubes.
Let's visualize the rectangular cube:
```
----------------------
/ _______ ______ ______/|
/ / / / /||
/ /______/______/_____/ ||
| | | | | ||
| | | | | |
| | | | | |
| |______|______|_____|/
|/______/______/______|
```
Each small cube that has been painted on exactly 3 faces can be represented as a corner cube that lies on the edge of the tunnel.
To determine the number of corner cubes, we need to count the number of edges along the tunnels and multiply by 2, as each edge has 2 corner cubes.
Since there are two straight tunnels of 4 cubes each, there are a total of 8 edge cubes:
```
____ ____ ____ ____
| | | | |
| | | | |
|____|____|____|____|
```
Now, let's multiply the number of corners cubes by 2 to account for both the top and bottom faces:
Number of corner cubes = 8 * 2 = 16
Therefore, there are 16 small cubes that have been painted on exactly 3 faces in this rectangular cube.
To determine the number of small cubes that have been painted on exactly three faces, we can break down the problem into smaller steps:
Step 1: Calculate the total number of small cubes in the rectangular cube.
In this case, the rectangular cube has sides measuring 4 cm, so the total number of small cubes within it would be:
Total number of small cubes = Length * Width * Height
= 4 cm * 4 cm * 4 cm
= 64 small cubes
So, there are 64 small cubes in the rectangular cube.
Step 2: Determine the number of small cubes used to create the tunnels.
There are two tunnels, each made with 4 small cubes. So, the total number of small cubes used to create the tunnels would be:
Total number of small cubes for tunnels = 2 tunnels * 4 small cubes per tunnel
= 8 small cubes
Step 3: Calculate the number of small cubes that have been painted on exactly three faces.
For a small cube to be painted on exactly three faces, it needs to be one of the corner cubes adjacent to the tunnels. Let's calculate the number of corner cubes first.
Number of corner cubes = 8 (since there are 8 corners in a rectangular cube)
However, we have already accounted for 4 corner cubes in each tunnel. So, we need to subtract those from the total.
Number of corner cubes without tunnels = 8 corner cubes - 2 tunnels * 4 corner cubes per tunnel
= 8 corner cubes - 8 corner cubes
= 0 corner cubes
Therefore, there are no corner cubes left to be painted on exactly three faces.
Step 4: Calculate the number of small cubes painted on exactly three faces.
Since there are no corner cubes left, the number of small cubes painted on exactly three faces would be 0.
So, no small cubes have been painted on exactly three faces.