Source
r/math•
JonMW

Self-posed problem: "Practical combat positioning for D&D Beholders" OR "how to efficiently move a cone to cover a given point"

Please forgive the large number of words. I am not great at precisely and concisely conveying mathematical concepts because I might use the wrong words for things.

This is a picture of a beholder. This is the conventional ingame stats of a regular beholder, for the 5th edition of D&D. The important rules and goals will now be covered:

  • The beholder and a single enemy are in combat.
  • Beholders create a cone (with themselves at the apex). The height of this cone is 150 feet, and the diameter of the base is 150 feet.
  • The beholder wants to keep their enemy inside the cone, because this makes the enemy weaker. The enemy will try to run out of the cone (though they cannot see the cone directly and will guess at the best way to run).
  • Beholders will shoot rays (pew! pew!) at their enemies. These travel in straight lines and must originate approximately from the beholder. The maximum range of a ray is 120 feet.
  • Rays are blocked by the cone. Using rays is the only way that the beholder can actually win the fight.
  • The beholder floats effortlessly through the air, and can move 20 feet at a time, in any direction.
  • The enemy cannot fly, and is limited to running on the flat ground. The exact running speed of the enemy is not known, but you can safely expect them to move 30 feet before shooting back.
  • The beholder can only change the direction of the cone at certain times - before they move and shoot. After the beholder moves and shoots, the enemy will have their turn, at which point they will try to exit the cone and shoot back.
  • Assume that the beholder's turn is first and he starts in a good position - though what that position exactly is, is still to be determined.

The goal, therefore, is for the beholder to aim the cone so that the enemy is barely outside it, shoot rays, and then move 20' so that the target will have the hardest time getting out of the cone - which one may interpret several ways (because the enemy may use many different strategies to try to exit the cone), but I interpret it to mean that the closest bit of floor outside the cone is as far away as possible from the target.

I have done a bit of thinking and research. I see how the angle of the cone in relation to the floor will create differently-shaped zones. Furthermore, I think that a strong strategy is to start very low to the ground, with the cone aimed just below the enemy - shoot, and then move directly upwards. That will produce a large, parabolic field, and at long ranges this may put the enemy more than 40 feet away from the edge (based on some back-of-a-napkin approximations) which is better than I had hoped. You couldn't do that turn after turn indefinitely because your angle gets worse with each movement upwards, but it's a good starting point.

However, I'm not sure how to begin attacking this in general to prove that this method is optimal, or how to modify it into a strategy that works indefinitely. Any guidance?

Posted May 15, 2019, 6:30 AM UTC

610
Moo-im-a-goat•

Just to be clear, you want every turn to be as follows: move cone, attack, move cone, move 20ft in every direction? Then the enemy (attacks and) moves 30ft?

EDIT any limitation in moving the cone?

2
JonMWOP•

Not quite. The turn is: Aim the cone, attack, move 20ft (in any direction).

The cone can be given its angle (i.e. bearing and pitch) only at the beginning, then that angle is locked in until next turn.

2
Moo-im-a-goat•

Aaah I see, I have to think again :) see, the system has rotational symmetry around the axis beholder/enemy (ignoring the ground for now), so you can think about the 3d situation with 2d images (which you then rotate) If what you want is to put the enemy as deep inside the cone as possible, it is actually the same as trying to find the biggest sphere that fits in your cone.

Thinking about a section of the cone (an isoceles triangle with base 150 and height 150), the biggest circle that fits is the inscribed circle, with radius approx. 45ft. So the best you can do is trap the enemy at least 45ft from any border of the cone.

optimal trap 1unit = 150ft

But I don't think you can achieve this configuration if you want to attack and move 20ft..

3
JonMWOP•

Mmm. Ok, I think I'm getting better at rephrasing this without all the game mechanics.

  • We have a point on the outer face of a cone of height and diameter 150ft. (cone shape)
  • The point must be within 120ft of the apex. (ray range)
  • The point is moved 20ft within the cone (relative movement)
  • A flat plane is placed to intersect the point (possible ground level movement)
  • What is the biggest circle that you can scribe on that plane with its center at the point, bounded by the surfaces of the cone?

My gut feeling is to place the point very far away. Let's say 100ft. Then move the point directly into the cone (move perpendicular to the cone face it's touching). The orientation of the plane, to maximise the size of the circle, however, is not so obvious.

3
jeff0•

Are you thinking of this strictly as a mathematical exercise or are you planning an encounter?

1
ReinH•

Probably not an actual encounter, because the actual D&D rules for cone targeting do not allow for this sort of thing. It's just a triangle with a point at the origin of the cone, as seen here. It could be house ruled, though, I guess.

1
JonMWOP•

Pre written module that includes a beholder, which I haven't used before. However, the arena is an odd one and has some qualities that make all this analysis impossible to apply usefully. That's fine, I still want to know.

1
jeff0•

Gotcha. I was thinking that a beholder might put a high-ceilinged defensive niche in its lair. That way it could cover the whole ground area of the niche with its cone, making an earth-bound opponent unable to leave the cone without breaking line-of-sight. But if it's in a pre-designed arena, that's moot.

Beholders are tricksy. My group recently faced one which had a stone wall that we could hide behind in its lair. Or so we thought, until it stone-to-fleshed it into a large slab of bologna.

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JonMWOP•

The actual arena completely inverts regular strategy. To keep things general, the floor lets walking creatures zoom straight out of the antimagic irrelevant of movement speed, and there's an Important Thing in a prominent position which the beholder does not want to hit with the cone which makes angling almost impossible.

My actual best strategy if I want to use the cone at all is to place the cone tangent to the enemy, then ready my action to dash and block them if they start to cast... which is easily defeated if they just move first. And I'm reduced to using legendary actions for fighting which cuts my DPS in half.

1
hh26•

Due to symmetry of up/down, you should be able to swap back and forth with this.

My first idea before I even read your strategy was for it to angle the cone just above the enemy, shoot, then move directly downwards, which results in the exact same advantage but requires movement downwards instead of upwards.

So it might be possible to do alternate an upward movement and a downward movement each turn and on average maintain the same altitude.

This could be mitigated by concerns about the Beholder either not wanting to get into melee range, OR actually wanting to get into melee range, since they can only cast three beams which are randomly chosen, so if they get a sleep or paralyzing one they'll have to bite to deal damage.

This could also be mitigated by the Beholder not being smart enough to actually think of this sort of sophisticated strategy. It might just shoot a beam and then move him into eye range horizontally because that's the simplest way to do this sort of thing, though not optimal.

1