A pair of opposite faces of a cube is painted yellow another pair of opposite faces orange and the remaining faces white. The cube is then cut in to 343 smaller but identical cubes.How many of the smaller cubes have exactly two colors on them?

To solve this problem, we need to understand the structure of the cube and how the colors are distributed.

First, let's visualize the cube. A cube has 6 faces, and each face has a pair of opposite faces. In this case, one pair is painted yellow, another pair is painted orange, and the remaining faces are painted white.

Next, we need to determine the number of smaller cubes that have exactly two colors on them. To do this, we can consider the different possibilities.

1. Smaller cubes with two yellow faces: Since there is only one pair of yellow faces, there can be a maximum of one smaller cube with two yellow faces.

2. Smaller cubes with two orange faces: Similarly, since there is only one pair of orange faces, there can be a maximum of one smaller cube with two orange faces.

3. Smaller cubes with one yellow face and one orange face: In this case, we need to consider the arrangement of the yellow and orange faces. Since there are two yellow faces and two orange faces, there can be a maximum of two smaller cubes with one yellow face and one orange face.

Therefore, the total number of smaller cubes with exactly two colors on them is 1 + 1 + 2 = 4.

In conclusion, there are 4 smaller cubes that have exactly two colors on them.