A quintomino is a regular pentagon having all its sides colored by five different colours. Two quintominoes are the same if they can be transformed into each other by a symmetry of the pentagon (that is, a cyclic rotation or a flip of the two faces). It is easy to see that there are exactly 12 different quintominoes. On the other hand, there are also exactly 12 pentagonal faces of a dodecahedron whence the puzzling question whether the 12 quintominoes can be joined together (colours mathching) into a dodecahedron.
According to the Dictionnaire de mathematiques recreatives this can be done and John Conway found 3 basic solutions in 1959. These 3 solutions can be found from the diagrams below, taken from page 921 of the reprinted Winning Ways for your Mathematical Plays (volume 4) where they are called the three quintominal dodecahedra giving the impression that there are just 3 solutions to the puzzle (up to symmetries of the dodecahedron). Here are the 3 Conway solutions

One projects the dodecahedron down from the top face which is supposed to be the quintomino where the five colours red (1), blue (2), yellow (3), green (4) and black(5) are ordered to form the quintomino of type A=12345. Using the other quintomino-codes it is then easy to work out how the quintominoes fit together to form a coloured dodecahedron.
In preparing to explain this puzzle for my geometry-101 course I spend a couple of hours working out a possible method to prove that these are indeed the only three solutions. The method is simple : take one or two of the bottom pentagons and fill then with mathching quintominoes, then these more or less fix all the other sides and usually one quickly runs into a contradiction.
However, along the way I found one case (see top picture) which seems to be a new quintominal dodecahedron. It can't be one of the three Conway-types as the central quintomino is of type F. Possibly I did something wrong (but what?) or there are just more solutions and Conway stopped after finding the first three of them…
Update (with help from Michel Van den Bergh) Here is an elegant way to construct 'new' solutions from existing ones, take a permutation $\\sigma \\in S_5$ permuting the five colours and look on the resulting colored dodecahedron (which again is a solution) for the (new) face of type A and project from it to get a new diagram. Probably the correct statement of the quintominal-dodecahedron-problem is : find all solutions up to symmetries of the dodecahedron _and_ permutations of the colours. Likely, the 3 Conway solutions represent the different orbits under this larger group action. Remains the problem : to which orbit belongs the top picture??

Conway, geometry, puzzle, symmetry