It's clear that I am not the only one for whom Martin Gardner provided important parts of growing up.
In my case, it was hexaflexagons. I happened on that copy of Scientific American with that column in it, and I was hooked on both Scientific American and hexaflexagons. I have subscribed continuously since then. I see that the article itself is so far back in the mists of time that Google provides it via one of Gardner's collections of his Scientific American columns.
I was at an age where I was fascinated by cootie catchers and other kinds of paper folding. The tetraflexagons had some resemblance to a cootie catcher, but they were much better. I never mastered the slight of hand necessary to convincingly change the no-cooties state to the cooties state, and, in any case, I liked more mathematical sorts of thing better. The hexaflexagons were extensible to numbers of configurations limited only by the thickness of paper and accuracy of construction. I think I got up to 24 faces on the biggest one I built.
I had been adept at mathematics from early on. I kept a notebook in which, during duller times in fourth grade, I would double numbers. I didn't know at the time that if I started with 1, I was enumerating the powers of 2. I did know that they were pretty. I was fortunate to live in an old house and to have tolerant parents who felt that writing on my bedroom walls and stapling, pasting, and taping my mathematical constructs to the walls and ceiling was a useful activity. I was also branching out from cootie catchers to the Platonic solids, via a number of library books, including W. W. Rouse Ball's Mathematical Recreations and Essays, a forerunner of Gardner's column and compilations, and Cundy and Rollett's Mathematical Models.
My family did not have a long academic pedigree; my mother was the first in her family to attend college and had a Master's in Library Science; my father was a high-school dropout. So I mostly had to find my own way into academia. Mathematics was satisfyingly lovely, but as I got into adolescence, I wanted to find The Meaning Of It All, which led me down the standard academic rabbit hole toward physics and philosophy.
Organic chemistry provided my conversion experience. It was the experiment, early in the course, on recrystallization. The quick formation of beautiful crystals hooked me as surely as hexaflexagons did earlier. As well as the nice little hexagons that I learned to draw quickly and neatly, as a chemist must.
It was that balance of the physical and the theoretical, geometry in the real world, that I really love, that my mind works well with. And Martin Gardner's article on hexaflexagons was the first time all that came together for me.
Good websites on flexagons and their construction can be found here and here. The image is from the first of those.