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No.
If you get multiple attacks because your base attack bonus is high enough, you must make the attacks in order from highest bonus to lowest. If you are using two weapons, you can strike with either weapon first. If you are using a double weapon, you can strike with either part of the weapon first.

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Out of curiosity, why would you want to do them out of order?
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I'm the DM in this case.
One of my NPC villains will have Deflect Arrows. (The NPC in question is a high level Rakshasa, thus making it a good candidate for an NPC wanting protection from ranged piercing weapons!)
One of the PCs in my group is an archer and, at present, has no other means of contributing in combat if his bow is useless.
I was wondering if the Rakshasa's ability to deflect arrows would keep the PC archer from contributing to the eventual fight? You know, he could hit the Rakshasa with his first attack reasonably well but his iterative attacks would have to be very high rolls on the d20 to hit. That could be unreasonably frustrating for the Player if he can do essentially nothing the whole fight.
I figured maybe if he used his iterative attacks first, then rolled his main attack, he could contribute to the fight.
....In any event, the combat in question is about four levels away so there's quite a bit of gaming between now and then.

Claxon |

I'm making some assumptions, but in general a PC should have a to hit high enough to almost guarantee that all hits but the last iterative will be successful (most of the time) except for on bad rolls (<8).
Also, a well built archer will have rapid shot and many shot.
That means the first shot has two arrows (unclear if deflect arrows should deflect 1 or both) and should also have many shot, which gives him an extra attack for free (rapid shot has -2 penalty).
It's worth noting that the extra attack from Manyshot isn't granted by BAB, and is technically excluded from the rule about when the attack must be taken (that iterative attacks must be done highest to lowest).
So he could do his Manyshot first, then the attack with Rapid Shot, then his two iterative attacks.
With BAB 11, and not factoring in other bonuses he would have attacks at:
11-2 (Manyshot 2 arrows of damage)
11-2 (Extra Attack from Rapid Shot)
6-2 Iterative
1-2 Iterative
Edit: Rereading Manyshot it specifies the first attack you make, so it seems you would have to do Manyshot on the first attack roll.
Since you are the GM, simply rule that Deflect Arrows only deflect 1 of the 2 arrows from Manyshot and he should be reasonably getting 3 arrows worth of damage each round (out of 5 total arrows worth of damage) possible.

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Deflect Arrows is one attack per round, and as Claxon said, an archer should have rapid shot at minimum, and most likely manyshot as well.
As a player, a NPC having deflect arrows is fair. I know I've used many times on m monks. It's not like your NPC is a flying manticore under a constant fickle winds spell (something I faced before. it was a nightmare). Archery is the strongest combat style in the game. Having a counter every once in a while is certainly valid, just as long as it's not every NPC.

KainPen |
only way to get around the highest to lowest is with extra attacks that are not caused because of your base attack bonus, such as haste other such spells, Swift actions(mostly mythic stuff but are a few exceptions Monk ki point), Speed weapons.
See this thread for more detail
http://paizo.com/threads/rzs2pyef?Clarification-on-use-of-Monk-Ki-pool#5

Bill Dunn |

You might consider foreshadowing the rakshasa's arrow deflection with another opponent the PCs might face. It's a fairly well known effect for people to plan ahead for their previous battles - get inconvenienced one time by some factor, you add a counter for that factor even if you have no guarantee of ever encountering it again. If you frustrate the archer now with an opponent with arrow deflection, then it's on him to make sure that he minimizes that problem in the future.

Artificial 20 |
It's minor, but Deflect Arrows does require a free hand to use. Perhaps you could engineer the fight so that itself would be a disadvantage to the NPC, like they're trying to hold onto a valuable McGuffin and if they put it down to start batting away arrows it's a lot easier for the PCs to retrieve (handling an AoO instead of trying to steal/disarm it off them).
Something like that could let the archer contribute through utility and harassment instead of just plain old HP depletion :).

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Deflect Arrows stops both arrows from multishot; doing differently is a house rule. But, it's a pretty tiny one, from a scope perspective: not many bad guys have deflect arrows.
But still, with haste and rapid shot, you've still got 3 attacks at a high bonus. Even if the iteratives all miss, he'll have two solid chances to hit.
Plus, if archer-man gets to go before the rakshasa, the baddie can't use deflect arrows, since he'll be flat-footed. Archers usually have pretty high initiative.