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Based on existing DCs, what would a ballpark number be for a group of 4 7th-level individuals to surf a single shared tower shield down a series of flights of stairs (each turning at 90 degrees at each corner) without stopping for over 150' of vertical distance, and what check or save would this be (I'm guessing Acrobatics, but there might be some canon for things like this to be a Dexterity Save or something else instead)?

A negative answer is perfectly acceptable - if there's nothing extant that is particularly applicable.

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Acrobatics (PHB 176 - emphasis mine)

Your Dexterity (Acrobatics) check covers your attempt to stay on your feet in a tricky situation, such as when you’re trying to run across a sheet of ice, balance on a tightrope, or stay upright on a rocking ship’s deck. The DM might also call for a Dexterity (Acrobatics) check to see if you can perform acrobatic stunts, including dives, rolls, somersaults, and flips.

Note that if all four of them are attempting to stay on a single shield, this would be a group check.

Group Checks When a number of individuals are trying to accomplish something as a group, the DM might ask for a group ability check. In such a situation, the characters who are skilled at a particular task help cover those who aren't. To make a group ability check, everyone in the group makes the ability check. If at least half the group succeeds, the whole group succeeds. Otherwise, the group fails. Group checks don't come up very often, and they're most useful when all the characters succeed or fail as a group.

Finally, see: What guidelines if any exist to guide the DM in establishing difficulty level?

You know more precise information that should factor into your decision - Polished marble steps? Unevenly worn stone? Carpeted? Are the stairs particularly steep or shallow? What is lighting like? etc. You also know the tone of your campaign - gritty realism or romantic fantasy?

Based on your description, I would likely assign this as somewhere between Moderate and Hard. Unlike the other two points, though, that is not a rule but rather a ruling, and it says as much about me as a DM as it does about the situation itself.

You should choose a number that reflects your assessment of how difficult this task is for these characters in this situation in your world.

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    \$\begingroup\$ Turning 90 degrees on a shield? Moderate looks too low for that. \$\endgroup\$ – Mołot Jun 2 at 23:42
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    \$\begingroup\$ What IS the DC to ollie into a wallride? \$\endgroup\$ – Darth Pseudonym Jun 3 at 3:55
  • \$\begingroup\$ @Mołot I agree - which is why I said it would be between Moderate and Hard - that is, above Moderate. But it would also depend on the length of the stairway preceding it and the size of the landing. Hence, it would require a DM ruling, not a rule. \$\endgroup\$ – Kirt Jun 3 at 5:29
  • \$\begingroup\$ @Kirt I would agree with Molot here but I want to explain out. Honestly I think this task should be impossible without magical aid. 4 people going down a set of stairs on a shield, totally fine. 4 people managing to turn a shield on a 90 degree turn. Well that looks a lot like a bobsled team. And no bobsled team has ever been given the task to make a 90 degree turn on a corner, because it wouldn't work. \$\endgroup\$ – GuidingOlive Jun 3 at 21:45
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    \$\begingroup\$ @GuidingOlive An actual 90 degree turn for the shield itself, at speed, would of course be impossible. I am assuming it is the staircase itself that is turning 90 degrees, and that the shield has the entire width of the landing to negotiate that. YMMV. I am at a loss as to how to say any more clearly that the DM sets the DC based on their assessment of the difficulty and that I am not suggesting that my assessment would play at your table. \$\endgroup\$ – Kirt Jun 3 at 22:18
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I would call this a very hard task, yet I'd want the party to have a reasonable chance of success if they are set up to be good at this kind of thing.

So, first, the relevant roll is Dex (Acrobatics) for all the characters, unless one of them can give you are really good case for something else.

I would probably have them pick their best guy for the "pilot", and he would have to make a DC20 check to run the shield. Each other character would need a DC15 check to stay on the shield, as a group check (2 out of 3; they all make it).

If they get a little smarter and use the Help action, I'd go with pilot DC20 with advantage from the helping guy, and the other two guys DC15 to stay on it; again group check (1 out of 2 keeps them both on).

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