Mini Cube
This puzzle is commercially available and is also known as the 1x2x2 Cuboid. It is so easy to solve that no sulotion strategy need to be given. The mathematical group underlying this puzzle is the symmetric group S_3.
Java Applet Instructions

The Green Flag resets the puzzle.

Pressing the Spacebar scrambles the puzzle.

Down Arrow turns the bottom two cubelets by 180 degrees.

Right Arrow turns the rightmost two cubelets by 180 degrees.

Left Arrow turns the leftmost two cubelets by 180 degrees.

Up Arrow turns the back two cubelets by 180 degrees.

To solve the prism, you have to get the top side yellow,
the left side blue and the front side red. (The green right side,
orange back side, and the white bottom are invisible.)

When you have solved the puzzle, your time taken is displayed
for 5 seconds.