Pack three circles in a triangle — is the obvious tightest fit actually the best?
Given a triangle, draw three circles inside it so each circle touches the other two and each also touches two of the triangle's sides, the natural way to pack three mutually-tangent circles into a corner-heavy shape. For over a century, this construction was assumed to be the arrangement covering the most possible area.
Reveal the answer
It never is: simply inscribing one large circle and filling the two remaining corners with smaller circles always covers more total area than Malfatti's mutually-tangent arrangement, for every triangle shape. Gian Francesco Malfatti posed the packing problem in 1803 assuming his construction was optimal, and it took mathematicians well over a century to rigorously establish that it never actually is.
— Gian Francesco Malfatti, Memoria sopra un problema stereotomico — 1803