# What does the distribution of a strictly increasing random walk look like if you stop at `N` steps?

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I have a puzzle inspired by the game Idle Brick Breaker (Google Play and App Store)

Say you have a d100, and if you roll a 100, you advance 50 steps (or `S`

), if you roll 1 to 19 (or `D`

), you advance 2 steps, and every other roll is 1 step.

After 1 million steps (or `N`

steps), or exceeding `N`

, you look back and see how many steps you "skipped" over in the process.

If you had the distribution of all possible ways you could roll, stopping after you reach or exceed 1 million, what is the mean of this distribution? What does it look like? How does it change with varying `S`

, `D`

, and `N`

?

Put another way, what does the distribution of a strictly increasing random walk look like if you stop at `N`

steps?

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