# Sunday Times Teaser 3166 – All Their Marbles

*by Peter Good*

#### Published Sunday May 28 2023 (link)

In their art class, Jack and Jill each had a box of spherical glass marbles. Each box contained marbles of at least three different sizes, and the radius of every marble was a whole number of cm. Jack placed three marbles of equal size from his box onto a desk, and placed each of the others in turn on top so that all four marbles touched each other and formed a triangular pyramid. He worked out the height of each pyramid (from desk to top of top marble) and obtained a different whole number of cm each time. Jill was also able to do this with her marbles but they were all of different sizes to Jack’s. None of the pyramids was higher than 30cm.

List all of the different marble radii in ascending order.

This is a somewhat surprising teaser since it turns out that there are six combinations of the three base marble radii (R) and the upper marble radii (r) that yield heights but only one of these is a true tetrahedron. In the other five, the tops of all four marbles are at the same height with the resulting ‘tetrahedra’ being no more than flat planes!