Watching a video about snakes and ladders (https://www.youtube.com/watch?v=k2ixp5VozIs) inspired me to dust off my markov-chain-memories and calculate the probability of winning the game after N rounds for normal and hardcore (ladders are snakes too) version.
Here’s my code: https://gist.github.com/SimonLammer/5f7c5fd4f9e60bba9fd13db0930ff83b

Normal: 61% after 55 rounds; 95% after 144 rounds; 99% after 233 rounds.
Hardcore: 4.5% after 55 rounds; 19% after 144 rounds; 32% after 233 rounds; 66% after 610 rounds; 95% after 1597 rounds; 99% after 2584 rounds.

I expected the hardcore version to be harder, but didn’t foresee a difference this big.

How could the number of expected snakes that were taken to win be calculated (aside from computer simulation)?