By “a local walk-in store”, I think he means readily-available, over-the-counter supplies.
METHOD 1
I’d suggest a square lantern battery wired to a bit of steel wool, but that might cause a little bit of molten steel (which would violate condition #2 in OP), and over-using the battery due to high current just might “over excite” it, resulting in a *ahem* “catastrophic failure” (which would violate condition #1 in OP). I’ve seen this done safely, and have done it safely myself, yet Scouting is Scouting, and we must follow the rules. (PS -- check with your Scoutmaster about this experiment, maybe he’ll approve it.)
METHOD 2
Justification for this method: You're near something (say a huge forest fire), and you need to start a backfire to stop the fire from advancing. Trouble is, the blaze is so hot you can't even get close enough to it to light a stick in your hand. You think to yourself, if you could just collect all that heat in one place to get it hot enough to start a fire. Hmm… then you remember a posting on a science forum … Here’s something along the lines of unique or never-thought-of methods. Use a magnifying glass to focus the energy from a campfire onto starter material, such as paper. Use a 3-inch glass -- bigger is better. Yeah yeah, I know, it doesn’t seems unusual to start a fire with a fire, but I think no one’s ever done it this way before.
METHOD 3
Justification for this method: The sun’s just set and you need to start a fire. All you have is the very warm side of a boulder or cliff (that’s facing west into the setting sun, of course) and a magnifying glass. You think to yourself, if you could just collect all that warmth in one place to get it hot enough to start a fire. Hmm… then you remember a posting on a science forum …
If Method 2 seems too weak or obvious, try this niftier method. Put a flat solid object (such as a metal plate, cast iron frying pan, slate, flagstone, etc) against the fire, and use
it as the source of heat energy to focus with the magnifying glass. When the stone becomes too hot to touch, it should be radiating enough energy to burn paper. I just invented this method, so I’m assuming this will work because we can sit in the sunlight okay and still start a fire with it, so a surface too hot to touch should definitely burn paper.
(Caution: Never put a stone in a river/water into or near a fire. Stones can become waterlogged over time, and when heated by the fire, the water can turn into steam and cause the stone to explode.)
Focal Length
For an approximation of focal length for Methods 2 and 3, use the focal length found when focusing light from the campfire. The image below shows that focal length is a function of wavelength, so the infrared focal length is even longer than that for red, thus, you’ll need to increase the focal length a bit to properly focus the
heat (not light) from the fire or hot object. I have no proof that this method will work, but it seems like it “should”. Try it out first before you decide to use it.