The marble problem asks for three nonoverlapping circles inside a given triangle with maximum total area. Malfatti (1803) posed it in terms of determining the three circular columns of marble of possibly different sizes which, when carved out of a right prism with a triangular base, would have the largest possible total cross section (Martin 1998, p. 92).
Malfatti proposed three circles (the Malfatti circles) tangent to each other and to two sides of the triangle. In 1930, it was shown that the Malfatti circles were not always the best solution. Then Goldberg (1967) showed that, even worse, they are never the best solution (Ogilvy 1990, pp. 145-147).
Zalgaller and Los' (1994) give analytic conditions for the solution of the marble problem. The solution is obtained by a greedy algorithm: first place the incircle of the triangle, then the largest circle that fits in the remaining region, and finally the largest circle that fits after the first two have been placed. Lombardi (2023) gives an analytic proof of the optimality of this construction.
