A body of mass (2M) splits into four masses m, M - m, m, M - m ,...

$U = \dfrac{G{m}^{2}}{a} + \dfrac{G ( M - m{)}^{2}}{a} + \dfrac{2Gm ( M - m )}{a} + \dfrac{2Gm ( M - m )}{\sqrt{2}a}$

For maxima or minima of potential energy $\dfrac{dU}{dm} = 0$

$\dfrac{dU}{dm} = \dfrac{G}{a} ( 2m - 2 ( M - m ) + ( M - 2m ) 2 + \sqrt{2} ( M - 2m )$

$( 4m - 4m - 2\sqrt{2}m ) + ( - 2M + 2M + \sqrt{2}M ) = 0; \sqrt{2}M = 2\sqrt{2}m; \dfrac{M}{m} = 2$