I lie on U

Note that the entire cube contains $9^3=729$ lattice points, and the interior contains $7^3=343$ lattice points. Thus, there are $9^3-7^3=\boxed{386}$ lattice points on the surface of the cube.

We classify the lattice points on the surface of $U$ into $3$ categories: the lattice points that are on a vertex of the cube, the lattice points that are on the edge of the cube but not on a vertex of the cube, and the lattice points that are on the surface of the cube but not on an edge of the cube.

There are $8$ vertices on a cube, and there is one lattice point per vertex, so there are $8$ lattice points on a vertex of the cube.

There are $12$ edges on a cube, and there are $7$ lattice points on each edge that is not on a vertex. Hence, there are $84$ such lattice points.

There are $6$ faces on a cube, and there are $7^2=49$ lattice points on each face that is not on a vertex. Hence, there are $(49)(6)=294$ such lattice points.

Adding them up all, there are $8+84+294=\boxed{386}$ lattice points on the surface of the cube.

We have 6 faces on the cube each one will have one constant coordinate and another two that can vary over the face. Each face can have 9 distinct values for the two coordinates that can vary so the amount of distinct pairs per face is 9(9) = 81.

We have 6 faces so we need to include these 6(81)=486 but we have now included each point that lies on an edge twice (excluding at vertices) and each point on a corner 3 times so we need to take away these extra points, the cube has 12 edges and 7 distinct points along each edge (excluding corners) so we need to take away 12(7)=84. So 486-84=402 and we also need to take the extra corners off. We have 8 corners and have included 2 extra points at each corner so we need to take away 8(2)=16. So the answer is 402-16=386. $386$

A key is to realize that from 0 to 8 (inclusive) there are 9 integers.

Therefore we have a total of 9x9x9 lattice points as part of the total VOLUME of the cube.

The easiest way to get the surface points only may be to subtract the 7x7x7 lattice points in the interior.

So we have 9x9x9 - 7x7x7 = 729 - 343 = 386

I started by drawing a 3-d diagram. Let us first count the points inside the square face but is not a side length of corner, that would give us a $7 \times 7$ square, of which there are 6 faces, so giving us $49 \times 6 = 294$ points.

We then count the number of points at the corner, which is simply the number of corners the cube has, which is $294+8=302$ .

Finally we count the edges. Each edge has $9-2=7$ points, and there are $12$ edges, thus giving us an additional $7 \times 12 = 84$ points, so our total is $294+8+81=\boxed{386}$

A cube has 6 faces, 12 edges and 8 vertices.

Each faces has 81 lattice points, but 4 of them are vertices, 28 of them are on edges. So each faces has only 81 - 4 - 28 = 49 lattice points for their own. So the amouth of lattice points that only lie on 1 face is 49 $\times$ 6 = 294.

Each edges has 7 points and there're 12 edges so the total lattice points on edges is 7 $\times$ 12 = 84.

There're 8 points on vertices.

So, in total, there're $294 + 84 + 8 = 386$ lactice points lie on the surface of U. That's the answer.