If a dice has 3 on the top, right back 6, right front 2, bottom 4, left front 1, left back 5 and is rotated by +180 about the z axis and +90 about the x axis, how many dots does each face have after the rotation?

To determine the number of dots on each face of the rotated dice, we need to apply the given rotations (180 degrees about the z-axis and 90 degrees about the x-axis) to the initial configuration of the dice.

Let's start with the initial configuration of the dice:

Top: 3
Right Back: 6
Right Front: 2
Bottom: 4
Left Front: 1
Left Back: 5

Now, let's apply the rotations:

1. 180 degrees rotation about the z-axis:
- The top face (3) remains unchanged.
- The right back face (6) becomes the left front face.
- The right front face (2) becomes the left back face.
- The bottom face (4) remains unchanged.
- The left front face (1) becomes the right back face.
- The left back face (5) becomes the right front face.

After the 180 degrees rotation about the z-axis, the configuration of the dice is as follows:

Top: 3
Right Back: 1
Right Front: 5
Bottom: 4
Left Front: 6
Left Back: 2

2. 90 degrees rotation about the x-axis:
- The top face (3) becomes the bottom face.
- The right back face (1) becomes the right front face.
- The right front face (5) remains unchanged.
- The bottom face (4) becomes the top face.
- The left front face (6) becomes the left back face.
- The left back face (2) remains unchanged.

After the 90 degrees rotation about the x-axis, the final configuration of the dice is as follows:

Top: 4
Right Back: 1
Right Front: 5
Bottom: 3
Left Front: 6
Left Back: 2

Therefore, the number of dots on each face of the dice after the given rotations are:
- Top: 4
- Right Back: 1
- Right Front: 5
- Bottom: 3
- Left Front: 6
- Left Back: 2