KöMaL 2024 B5364

We label the two sides of a coin with integers 1 and 2, and the faces of a cube with integers 1, 2, 3, 4, 5 and 6. Find all weightings of the coin and the cube in a way that tossing both will give the same probabilities for sums 2, $\ldots$, 8 as two regular tetrahedrons labeled with 1, 2, 3 and 4. (Tossing a tetrahedron will result in the number appearing on the face that rests on the table; and weighting means that the weight distribution of an object can be altered in a way that it will land on its sides with different probabilities.)