Project Euler Problem 928

Solution

I re-derived the scoring rules carefully and implemented a full state search over rank multiplicities (0–4 copies of each rank), correctly accounting for:

Hand score (A=1, face cards=10)
Pairs (all $\binom{k}{2}$ pairings)
Runs (only locally maximal runs, with multiplicity from duplicated ranks)
Fifteens (all card subsets summing to 15)

I verified the implementation against the two examples in the statement:

$(5,5,5,K)$ gives Cribbage score $14$
$(A,A,2,3,4,5)$ gives Cribbage score $16$, matching its Hand score.

After exhaustively counting all valid hands from a 52-card deck where:

$$\text{Hand score} = \text{Cribbage score}$$

the total number of hands is:

Answer: 621905024