Start by solving the corners using the instructions for the 3-color cube, to do that just think of the color on each pair of opposite faces as being the same (red-orange, yellow-white, blue-green), if that seems unintuitive then look at this. The following examples show what a 3-color solution looks like on a 6-color cube, corner facelets match either the center color or that of the opposite face.

After solving the corners as a 3-color cube there are three possible states that can be reached using half-twists only:

1) The corners can be solved on all sides

2) The corners can be solved on one pair of opposite faces only

3) No faces can be solved

The following table shows how to tell which opposite face pairs are solvable with half-twists, a more detailed description is here.

If only one pair of opposite faces can be solved then place the cube so the solved faces are on the left & right to make the setup for the Waterwheel Sequence which is shown below. For this setup it is only necessary that the left & right faces are solved, the other faces can be ignored (do not have to match cube below). The sequence does not solve the corners, it makes them solvable with half-twists.

After applying the sequence (press Play) you can see in the above table (2-2 split, diagonal) that the corners can be solved using half-twists. Go ahead and complete the solution using the mouse. If it is not immediately obvious how to solve the corners with half-twists then look at this.

The first four moves of this sequence make the Waterwheel configuration. The fifth move is the one that "fixes" the 6-color configuration so it can be solved using half-twists. The last four moves are the inverse of the the first four. You can see how the sequence works on all possible configurations here.

If no faces can be solved then the above sequence will fix one pair of opposite faces so that the corners can be solved with half-twists, no setup is needed. To complete the solution, apply the sequence again as described above. To avoid having to do the sequence twice use the Parallel Sequence instead.

The 6-color corners can be solved before or after the 3-color edges are solved. If the 6-color corners are solved first then solving the 3-color edges slightly scrambles them but they remain solvable with half-twists. It is not actually necessary to solve the corners prior to solving the edges, just making them solvable with half-twists is sufficient. If the 3-color edges are solved first then one minor additional step is needed as shown here.

For random corner cubes that are already solved as 3-color, the probability for the number of face pairs that are solvable (as 6-color) with Half-Twists Only (HTO) is shown in the following table along with the sequence used for each.

You can try solving random corner cubes with half-twists here by selecting the option: "6-Color Corners: Pre-Solved as 3-Color (HTO)". There is also a similar option "ALL" for which the majority of configurations will require use of one of the above sequences to make solvable with half-twists.