Seven Bridges of Koenigsberg

I remember reading about this problem and Euler's solution when I was about 20. I believe it was Eric Temple Bell's Men of Math. It was my introduction to topology. Somewhere I got a translation of his paper. It was a nice, clean, and clear solution to something that was mathematical but didn't start with equations. Very accessible.

Although in a different area, Archimedes' Sand Reckoner is also accessible and makes comprehensible some limitations of the ancient numbering system.