Probability to win Snakes&Ladders after n rounds

snakes_and_ladders.sage

snakes = {
    17: 7,
    54: 34,
    62: 19,
    64: 60,
    87: 24,
    93: 73,
    95: 75,
    99: 78,
}
ladders = {
    4: 14,
    9: 31,
    20: 38,
    25: 84,
    40: 59,
    51: 67,
    63: 81,
    71: 91,
}
if False: # ladders are snakes too
    snakes |= {v:k for k,v in ladders.items()}
    ladders = {}
print(f"{snakes=}")
print(f"{ladders=}")
board_size = 100
die_min = 1
die_max = 6
m = matrix(QQ, board_size, board_size)
die_probability = 1/(die_max-die_min+1)
for i in range(board_size-1):
    for j in range(die_min, die_max+1):
        destination = i+j
        if destination >= board_size: # overshoot back into the board
            destination = board_size-(destination-board_size)-1
        m[i,destination] += die_probability
m[board_size-1,board_size-1] = 1 # Stay at finish once reached
for transportations in [snakes, ladders]:
    for start, destination in transportations.items():
        for i in range(board_size):
            m[i,destination-1] += m[i,start-1]
            m[i,start-1] = 0

start = vector(QQ, board_size)
start[0] = 1
for i in (1, 2, 3, 5, 8, 13, 21, 30, 34, 55, 89, 100, 144, 233, 377, 610, 987, 1597, 2584):
    state = (start * m ** i)
    #print(f"{i:4d}: {state.numerical_approx(digits=2)}")
    print(f"{i:4d}: {state[board_size-1].numerical_approx(digits=2)}")

I was inspired to calculate this by https://www.youtube.com/watch?v=k2ixp5VozIs