Slim Tower - Solution
The following algorithms are taken from the results of a computer program that I did to search all possible positions. Since there are only 241920 positions on the slim tower, it isn't too tricky for a computer to index all of them along with the algorithms that generate and solve each position. The algorithms should be pretty much optimal for the positions described. The numbers in brackets at the end of each algorithm are the number of face or slice turns.
When following this solution, you hold the slim tower as you see in the image below, a 2x2 face is on the top and bottom.
U and D are used to represent turns of the top and bottom faces - just like the cube. F and R are double turns of these faces - a single turn isn't possible on the slim tower. E is a twist of the layer between the top and bottom layers turned in the direction of U.
Stage 1 - Bring The Corners Into Their Correct Layers
In this stage you make sure that the top and bottom faces show only one colour each. On my cube these colours are black and grey. So I have to make it so that the bottom face is totally grey and the top face totally black. You might be able to do this intuitively with combinations of R, U and D turns. You don't need to position them straight away - just look to group them into layers.
The diagrams of the bottom face are what you would see if you could look straight through the top two layers.
Stage 2 - Permute The Corners In Their Layers
In this step, you put the corners into their correct positions. The diagrams below show you a variety of ways of swapping corners about in a layer.
Stage 3 - Solve The Middle Layer
Swap the centre pieces around using the following algorithms.
The first algorithm is an interesting one. It swaps the edges on the middle layer of the right face. On a cube, it would also swap the front and back edges of the top layer. I use this on the domino to do just that.