You have to take the 1/focal length rule with a pinch of salt when dealing with a compact digicam. Let's forget about the focal lengths and crop factors for a moment and take a step back and look at what's happening.
You can't hold any camera perfectly still, that's just a fact of life, so as a result the lens is waggling about (albeit small amounts) while you frame the shot. That waggling is over a certain angle range so as you increase the focal length that waggling becomes more apparent due to the reduced angle of view. Now if you use a slower shutter speed the waggling causes camera shake (or blurring as we're calling it), the longer the focal length the more obvious the shake because the waggling angle is a bigger proportion of the angle of view.
So how do we reduce shake? You need to use a shutter speed which effectively "stops the action" in the same way as you'd want a higher shutter speed to freeze a footballer heading a ball for example. At longer focal lengths the waggling is more pronounced - the action is faster - so you need a higher shutter speed than at shorter focal lengths.
On 35mm systems trial and error has determined that for most situations, most people and most cameras that a shutter speed of 1/focal length is enough to prevent camera shake. With good technique etc that can be reduced hence it's only a rule of thumb.
For compacts it's a bit of a different ball game. The focal lengths are shorter but the posture for holding the camera is different, you don't really have the ability to brace elbows and squash the camera against your face to stabalise it. Therefore the 1/focal length rule doesn't necessarily apply. However, if you use the 35mm equivalent focal length then you do get a 1/focal length speed which you can have a good chance of hand holding.
