IBM Ponder This #218 · June 2016

This month's challenge is from Bert Dobbelaere. (Thanks!)
The marketing boys behind a large beverage brand decide to take advantage of a much-hyped soccer tournament by changing the shape of their plastic canisters. The new canisters are shaped as a truncated icosahedrons (achieved by taking an icosahedron and cutting off one-third of each edge on both ends; see http://mathworld.wolfram.com/TruncatedIcosahedron.html), which are often associated with the sport. The canisters have a small tap just above the bottom face. A pentagon was chosen as bottom, as this accentuates the symmetry of the shape. There were some concerns about the stability of the canister, which is typically placed near the edge of a table top.

On the assumption that the empty weight of the canister is negligible and its walls are infinitely thin, what is the maximum tilt angle of the table, so that the canister will never lose its balance, independent of the direction of the tilt and the filling level of the canister? If the maximum tilt angle is marginally exceeded and the canister starts to tumble, what fraction of its volume is filled with fluid?

Solution

Best opened after a real attempt

To be added.