Enigma 104: Cube balancing odds
From New Scientist #1248, 9th April 1981 [link]
Alf was showing his friend Maisie Moshan a uniform cube, like the one in the sketch, which had sockets at each vertex, mid-edge and mid-face, in each of which a little gold ball could be lodged.
“What I wonder,” he said, “is if I insert two little balls at random, what is the probability that I can then insert a third at a place which will leave the cube properly balanced?”
Maisie pondered. “By ‘properly balanced’ you mean with its centre of gravity still at the centre of the cube?”
“My guess,” said Maisie, “is that you would improve your chances if, instead of inserting the first at random, you put it at a mid-edge. Then you place the second at random, and the third at your choice, as before.”
Was Maisie right? What is the probability of ending up with a properly balanced cube, with three added balls, under (a) Alf’s system; (b) Maisie’s system?
Enigma 4 was also about balancing a cube.