High cadence is the easiest on your legs and knees, it's also the most efficient way which is why you're quicker. You can increase your strength by keeping the same cadence but using a higher gear ... but do it in bits, a cheap form of interval training.
Depending on the cadence you normally run, you can get significant improvements by working at higher cadences. Back when I rode with gears, I'd climb in the mid 90's or higher on a low gear.
Low cadences and high gears aren't the best for your legs and knees - this is the one drawback of my fixed gear fetish, and it has taken some very careful management of my body to avoid hurting something. Build your strength slowly and effectively, don't try to rush it or you'll lose the gains while the sore bits recover.
I rode my pb up norton a few weeks back in 14:06 and recorded a cadence of 81. I train with a power meter and its great to prove to yourself what gears work best for you. Best way to get faster is to ride with faster riders Ive found. We meet at 7pm at the tower hotel and at 6:10am. Both rides contain some of the fastest climbers in oz. Some are vets and some are recent national road champs. All are welcome to join us. Just jump on the back and grit your teeth! :)
What day is the 6.10am ride on??
I find when I run out of steam at the higher cadence & start dropping speed, I drop a gear and can pick it up again. Standing briefly also helps rest the group of muscles just used, but again, I find at the expense of speed.
The more you do it & the fitter you get, the longer you can hold the optimum cadence....
Conventional wisdom says to increase your average cadence and spin up the hills. As you get fitter you'll find you can maintain that cadence in the next harder gear, and then the next.. and so you'll improve as Richard said.
But as an experiment last year, for about a month I did my usual climb to the freeway bollards on the big ring (usually twice a week). It probably didn't do my chain any good crossing at 53-25, and I wouldn't have tried it if I felt any stress on the knees, but I did feel a lot stronger when I switched back to high cadence riding at the end of it..
edit- just read durian's post which is spot on. Chasing anyone a bit faster than you is the best way to improve- forget the numbers, just suck it in a chase hard!
You are right Don, when you switch back to your lower gear you will be stronger. It's like weight lifting for your legs. Slower reps. with a heavy weight. But, to try to better your PB each time out is "according to me" not a good idea. Pick a hill with a reasonable gradient but not to steep, that takes over ten minutes, stay in the saddle and ride reasonably hard, in a big gear.
This should be bigger than is comfortable for you.. Gradually incresae your reps. like in the gym. I think you blokes are talking about training, and not going for a ride?
I should add that this has to be balanced with pedaling at a high cadence down hills as well, try to eliminate free wheeling as much as possible. I know all this comes at a cost of not beating your mates on a week day, we have to make sacrifices for a postive gain.
What is really nice about riding fixed, is the mad sprint down the other side - you're forced to work at a much higher cadence than you would on something that allows you to coast and those extreme revs clear the lactic acid out of the muscles - this is a known 'active recovery' method. My ride to work involves going down the Veloway Hill but also has sections that work the legs hard - my legs always feel worse during the day at work than they do after the ride home, which involves climbing up that *&^%$ hill.
Both is best, people who have one without the other can benefit from working on their weaknesses.