There's no reason you need the centre of the lens to start a fire. I once had a Fresnel lens that broke with repeated handling - I took the small piece that broke off and was able to focus the sun's rays with it. (However, it was much too small to be able to start a fire.)

You can easily verify this (I haven't, btw, but I'm confident it would work) by covering most of the lens with a sheet of cardboard, or putting it in a manila envelope so only part of the lens protruded.

The main reason people think you can't start a fire with a "cut-down" fresnel lens is probably one of the following three:

1. They've never tried it.

2. They're using a fragment that is too small. You need fairly bright sunlight to start a fire with a credit-card size lens, especially one that's been knocking around in your wallet and isn't as sharp as when you bought it.

3. They fail to realise that the focal point doesn't move. If you take an 8.5 x 11 sheet and cut it into 4 pieces, each of the pieces should be able to start a fire in decent sunlight, but the focal point in each case will be exactly where it was before - i.e. it will be at the centre of the original sheet, which puts it at the corner of the cut-down sheet. For example, if you cut a credit-card size piece from the corner of an 8.5 x 11 fresnel sheet, the focal point (i.e. the point of light that the smoke comes out of) will be offset by about 6 inches from where you would normally expect it to be. But it will focus light, and it should be enough to start a fire if your fragment is big enough.

The only advantage I can see to this is that you could carry four credit-card sized pieces that could be fitted together to make a bigger lens. Each piece individually could start a fire in bright sunlight, even if you lost the other three. Otoh, if you have three of the four pieces, and you taped them together, you would probably be able to start a fire more quickly or in slightly worse conditions (e.g. winter, or later in the day, when the sun isn't as strong.)