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This question is just for fun and I apologise if it's too broad or off topic.

The Details:

As anyone who has played the Tony Hawk games can tell you, skateboarding - at least in part - can be scored using $\Bbb N$. Somewhat arbitrary points can be awarded for each successful trick in proportion to its perceived difficulty. These tricks can form sets. For example:

  • $\square$ for flip tricks,
  • $\bigcirc$ for grab tricks,
  • $\triangle$ for grinds,
  • $\updownarrow$ for flat ground tricks

The flip and grab tricks can be done with frontside $(+)$ or backside $(-)$ rotations in intervals of $180°$, multiplying the score of the trick by some factor. The grinds and the flat ground tricks are each done for a duration; the longer the duration, the higher the score; and the same is true for grab tricks

Skateboarding is sequence of combinations of tricks, such as

$$\text{frontside }180\text{ kickflip}\to 3\text{ second switch frontside fifty-fifty}\to 1\text{ second switch indy grab}.$$

It's also impossible to do certain trick combinations. For instance, you can't do a nollie out of a manual.

The Questions:

What kind of mathematics describes the set of all skateboarding combinations (and the point-scoring system)? Do any mathematical phenomena model the same mathematics?

Thoughts:

Surely there's at least one coherent answer since the Tony Hawk games exist.

I guessing a language in $\square\cup\bigcirc\cup\triangle\cup \updownarrow\cup (0, \infty)\cup\{\pm 180n°\mid n\in\Bbb N\}$ might suffice for the first question, with some function from the language to $\Bbb N$ for the scoring system.

The second question beats me.

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    $\begingroup$ The combinations would be described in a language you describe, or possibly an array recording the order and number and quality of each kind of trick, while a scoring system is just a map from the arrays $A$ to an interval e.g. $F:A\rightarrow [0,\infty)$. $\endgroup$ – Nij Aug 13 '17 at 0:45
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    $\begingroup$ All the TH games have a system in place where the base score for a trick lowers for each use (up to four times, I think). Going to the Switch stance adds $20\%$ to the base score and similarly deavlues from there. Not sure what else you want for scoring. $\endgroup$ – Sean Roberson Aug 13 '17 at 4:10
  • $\begingroup$ Also, see here for maybe an explanation of how scoring works in THPS3, using PAL frame counts: gamefaqs.com/ps2/476948-tony-hawks-pro-skater-3/faqs/18941 $\endgroup$ – Sean Roberson Aug 13 '17 at 4:10
  • $\begingroup$ I don't know much about skateboarding, but I assume there is a directed graph with simplified states (height and rotation status and velocity?) and transitions given by pieces of moves (and the action of gravity). If you look up "juggling state graph" or similar you can find examples showing possible juggling patterns. $\endgroup$ – Mark S. Aug 21 '17 at 14:49
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    $\begingroup$ +1 for formalizing a sport activity!... I sense the at least one rotation orientation should be a patameter for these operators. Some path integral should be added in order to evaluate curvature difficulty over flat and possibly nonflat surfaces. And i dont see tricks jumping and sliding over angled surfaces with the flat part of the table. These of course not knowing about the game, but appreciated in the sport... $\endgroup$ – Brethlosze Aug 21 '17 at 16:47

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