Rolling the Dirac dice 3 times results in 27 quantum universes. However, the dice total is
one of only 7 possible values. Instead of handling 27 values, we encode the possible dice
totals with the number of times that they occur. For example, a score of 3 (1 + 1 + 1) only
happens once in the 27 rolls, but a score of 6 happens a total of 7 times.
The initial deterministic dice roll total is 6 (1 + 2 + 3) and increases by 9 each turn.
An interesting observation is that since the player’s position is always modulo 10, we can
also increase the dice total modulo 10, as (a + b) % 10 = (a % 10) + (b % 10).
Memoization is the key to solving part two in a
reasonable time. For each possible starting universe we record the number of winning and losing
recursive universes so that we can reuse the result and avoid unnecessary calculations.