May 2024 - Solution
May 2024 Solution:
The numerical solutions are: * Main problem: 422378820864000 * Bonus: 90399509839271079668491458784005740889517921781547218950513473999637402251071324160000000000000000
There are several approaches to find those numbers; we present only one.
A Dobble-like deck corresponding to a parameter N is equivalent to a **projective plane** of order N. This is a set of N^2+N+1 points and N^2+N+1 lines such that every line contains exactly N+1 points, and every two lines intersect at exactly one point (think of a projective plane as a generalization of the Euclidean plane, with the difference that there are no parallel lines).
There is only one projective plane of order 4 and only one of order 8, up to isomorphism. Since we consider isomorphic decks as distinct if there are cards which differ in their symbols, given a Dobble deck we also need to count all the distinct Dobble decks obtained from it by taking a perumtation of the N^2+N+1 symbols in the deck. If the permutation is an **automorphism** of the deck, we'll double-count the resulting deck, so the solution is obtained by dividing (N^2+N+1)! by the size of the automorphism group for the corresponding projective plane, which for the relevant cases are 120960 for N=4 and 49448448 for N=8 (these numbers are well known, but can also be computed via a software package).