First, I think it's important to understand how safety matches work.
The Bottom Line
The primary flammable ingredient in a safety match head is sulfur, which ignites at at around 230 degrees celcius (about 450 farenheit), which is about the same as a piece of paper and a bit lower than firewood. If you can heat the match head to 230 C, even briefly, you're set. Doesn't matter how you get there.
The Normal Way
To achieve the required temperature, normally the match head is scratched across the striking surface. The red phosphorus in the striking surface combines with the potassium chlorate in the match head, and the friction from scratching generates enough heat to ignite the resulting mixture, which generates sufficient heat to ignite the sulfur. You can read more about safety matches here.
As the OP has discovered, a soggy or overused match box still has the right chemicals for the initial reaction, but its surface can no longer generate the abrasion required to mix the chemicals or the friction to heat the mixture. So it's very important to keep that match box dry!
The following are some alternative ways to get that match to ignite.
Use Any Available Heat Source
Anything will work as long as it gets hot enough to cook with: the coil burner of an electric stove, a frying pan or steam iron that's still hot, stored embers from a campfire or charcoal from a grill, car exhaust, a magnifying lens in daylight, etc. Or any existing flame (obviously).
If you can move an electric current through a sufficiently resistant circuit, that resistance can generate significant heat. This is how things like incandescent light bulbs and toasters work.
There are many guides online, such as this one. But the gist of it is the same:
- Use a long, thin piece of metal as a wire to connect the two terminals of the battery.
- Touch the match head to the wire.
- Wait for the wire to get hot enough to ignite the match head.
- The battery doesn't have to be too strong. In fact, it's probably safer to use a weaker battery that isn't strong enough to zap you. This method is known to work with a 9V battery, a couple of AAs, or even a smart phone battery.
- Some methods will instruct you to damage or destroy the battery, but this is not necessary.
- For the conductive material, you can use plain wire or all sorts of things: aluminum foil, a paperclip, a staple, some steel wool, graphite from a pencil, whatever.
There are variations on this method. One method runs an electric current through a gum wrapper until the wrapper itself ignites - no match required. Another method uses a short length of #2 pencil, some jumper cables, and a car battery. If the battery is strong enough to emit sparks, those sparks can also be used to ignite nearby combustibles.
Friction generates heat. More friction, more heat. Find a rough yet fine-grained surface and rub the match head over it until it gets hot enough. Concrete, sandpaper, the bottom of most home-made ceramics, and certain types of stone (if dry) all work. The trick here is reaching the required temperature before grinding all of the sulfur coating off the match head.
One trick that has been used is to grind match heads against other match heads. Popular Mechanics demonstrates a good way to do this. You'll need five matches.
- Hold four matches together tightly in a square bundle, match heads all together.
- Take the fifth match and press its head firmly against the heads of the other four.
- Twist your hands sharply in opposite directions, grinding the match heads against one another.
There are lots of great fire-starting tools and methods out there other than matches and fuel-based lighters. Tinderboxes. Flint and steel. Swedish firesteel. Electric lighters. All have their pros and cons. For backpacking, I recommend carrying at least two different firestarting methods. That way if your matches get wet or your lighter leaks itself dry, you'll still have another way to cook your food and stay warm.
And remember, always practice fire safety!