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Speedrun
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==Practicing==

==Practicing==

−Some games once you have enough speedrunning skill can probably be treated by a random walk for how far into the game you are after each amount of time. Suppose a game can be treated like a 1,000,000 step random walk starting from 0 and the speedrun can be divided up into 1,000,000 parts and every part has exactly 2 possible lengths of time that differ by the same amount and being faster in a part is considered moving up 1 in the random walk and being slower in a part is considered moving down 1. After 1,000,000 steps, the probability function with domain even numbers of where you end up is approximately given by e<sup>1/2,000,000 x<sup>2</sup></sup> times some constant. A normal distribution is any distribution that can be gotten from that one by stretching it along the x-axis by a positive amount, stretching it along the y-axis by the reciprocal of that amount, and moving it along the x-axis any amount. It can be proven that the [https://en.wikipedia.org/wiki/Standard_deviation#Definition_of_population_values standard deviation] of the function e<sup>-1/2x<sup>2</sup></sup> is 1. If your current record corresponds to a 1,000,000 step random walk that ends at 20,000, then the [https://en.wikipedia.org/wiki/Expected_value expected number] for your next record is 20,050. However, it is not worth resetting if you're at 9,500 after 500,000 steps. The math shows that given that a 1,000,000 step random walk ends at 20,000, the probability distribution of where you were half way through the random walk is normal with a standard deviation of 500. Although it's so much rarer to advance 10,500 steps in the second half than 10,000, it's also so much less rare to advance 9,500 steps in the first half than 10,000 that the probability of advancing 9,500 steps in the first half then 10,500 in the second half is only square root of e times less than the probability of advancing 10,000 steps in the first half and 10,000 in the second half~~. A 1,000,000 step random walk that ends at 20,000 doesn't have that great a chance of not taking you to the number 50 less than you were at after the same number of steps in your current record any time during the random walk so if your record is 20,000, you're just multiplying the expected number of attempts before your next record by a large number if you reset every time you are 50 behind where you were after the same number of steps in your current record~~.

+Some games once you have enough speedrunning skill can probably be treated by a random walk for how far into the game you are after each amount of time. Suppose a game can be treated like a 1,000,000 step random walk starting from 0 and the speedrun can be divided up into 1,000,000 parts and every part has exactly 2 possible lengths of time that differ by the same amount and being faster in a part is considered moving up 1 in the random walk and being slower in a part is considered moving down 1. After 1,000,000 steps, the probability function with domain even numbers of where you end up is approximately given by e<sup>1/2,000,000 x<sup>2</sup></sup> times some constant. A normal distribution is any distribution that can be gotten from that one by stretching it along the x-axis by a positive amount, stretching it along the y-axis by the reciprocal of that amount, and moving it along the x-axis any amount. It can be proven that the [https://en.wikipedia.org/wiki/Standard_deviation#Definition_of_population_values standard deviation] of the function e<sup>-1/2x<sup>2</sup></sup> is 1. If your current record corresponds to a 1,000,000 step random walk that ends at 20,000, then the [https://en.wikipedia.org/wiki/Expected_value expected number] for your next record is 20,050. However, it is not worth resetting if you're at 9,500 after 500,000 steps. The math shows that given that a 1,000,000 step random walk ends at 20,000, the probability distribution of where you were half way through the random walk is normal with a standard deviation of 500. Although it's so much rarer to advance 10,500 steps in the second half than 10,000, it's also so much less rare to advance 9,500 steps in the first half than 10,000 that the probability of advancing 9,500 steps in the first half then 10,500 in the second half is only square root of e times less than the probability of advancing 10,000 steps in the first half and 10,000 in the second half.

Also, if you still have more skill to be gained, you will probably gain it faster if you do no reset runs because every time you reset after failing a trick and costing a lot of time, you miss your chance to get practice on all those later hard tricks and slow down your ability to gain experience from which you can figure out what to try and see what happens. Probably almost everybody can keep taking in new information and retaining it but at a very sluggish rate. If that turns out to be the case, one thing you could try is to learn how to write a formal proof in a weak system of pure number theory. You could try mentally figuring out proofs of statements and adding at a sluggish rate, adding them to the list of statements you have proven and retained and figuring out new statements from ones you recalled from the list and occasionally adding them to that list. You could also slowly add to your mental list statements about your past speedrunning experience and keep adding more statements some of which can be figured out from statements you previously figured out and some of which can be gotten by combining a new observation with a statement you previously added to your list.

Also, if you still have more skill to be gained, you will probably gain it faster if you do no reset runs because every time you reset after failing a trick and costing a lot of time, you miss your chance to get practice on all those later hard tricks and slow down your ability to gain experience from which you can figure out what to try and see what happens. Probably almost everybody can keep taking in new information and retaining it but at a very sluggish rate. If that turns out to be the case, one thing you could try is to learn how to write a formal proof in a weak system of pure number theory. You could try mentally figuring out proofs of statements and adding at a sluggish rate, adding them to the list of statements you have proven and retained and figuring out new statements from ones you recalled from the list and occasionally adding them to that list. You could also slowly add to your mental list statements about your past speedrunning experience and keep adding more statements some of which can be figured out from statements you previously figured out and some of which can be gotten by combining a new observation with a statement you previously added to your list.

16

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