PIERCER You have achieved a penetrating precision in com- bat, granting you the following benefits: • Increase your Strength or Dexterity by 1, to a maximum of 20. • Once per turn, when you hit a creature with an attack that deals piercing damage, you can reroll one of the attack's damage dice, and you must use the new roll. • When you score a critical hit that deals piercing damage to a creature, you can roll one additional damage die when determining the extra piercing damage the target takes.
some observations I’ve made is that while the 2nd and 3rd bullet requirea an attack roll that deals piercing damage to be used, the wording doesn’t seem to require the the reroll or additional crit die to be piercing damage. It seems to be worded in a way that allows any damage die to be rerolled or chosen for a crit so long as a portion of the damage is piercing damage.
this got me thinking about smites, sneak attacks, and the weapon cantrips.
so my question is, can anyone explain how the single reroll mechanic might be quantified when more die are introduced? I don’t know if my guess is wrong or not but I suspect that when you add more die to a roll, and pick the lowest roll to reroll, that it might have more of an impact on the DPR than im visualizing.
for example, a level 5 high elf rogue who has chosen the booming blade cantrip would potentially deal 2d8+3d6+mod when casting that cantrip with with a rapier and sneak attack conditions met. How much damage increase is actually gained from cherry-picking the lowest roll to reroll between the 5 dice? or 11 dice on a crit…
I think the third bullet requires the extra die to be from the piercing damage for sure: “one additional damage die when determining the extra piercing damage”.
As for the reroll, well, I don’t know the stats on it but I’d imagine that the more dice you roll, the more likely you are to select a 1 or 2 to reroll, but overall you aren’t likely to add much more than half a die’s worth of damage no matter what.
I think the third bullet requires the extra die to be from the piercing damage for sure: “one additional damage die when determining the extra piercing damage”.
As for the reroll, well, I don’t know the stats on it but I’d imagine that the more dice you roll, the more likely you are to select a 1 or 2 to reroll, but overall you aren’t likely to add much more than half a die’s worth of damage no matter what.
If you only ever reroll a die that shows below average, but then roll average on the reroll, then you’d add about 1.5 for a d6, about 2 for a d8, and about 3 for a d12, I think. From there you need to consider that you have more chances to roll a die under the average if you have more dice.
Thank you. Do you know if there’s a way to quantify the likelihood of rolling “low” on at least 1 die given that the die can be of differing types and quantities?
You know basic Probability calculations, right? Getting a 1 on a d6 is 1 in 6, or 16.6 →%. Getting a low die roll is 50% if you consider low to mean less than a die's average. Getting a 1 on each die of a 2d6 roll is 1/36. Getting a 1 on either of a d6 or d8 is 4/24+3/24 = 7/24 = about 29%. So on and so forth.
Thank you. Do you know if there’s a way to quantify the likelihood of rolling “low” on at least 1 die given that the die can be of differing types and quantities?
If that's all you want, the probability of a single die rolling low is 50%, regardless of number of sides. The probability of at least one die being low in "d" dice is 1-(1/2)^d.
Substantially more complicated is when you care about how low the low roll is. That's where number of sides becomes relevant; a d12 is much more likely to roll much less than its own average than a d4 is. If you want to experience this in practice, do some test rolls applying Piercer to a dagger and a musket. The musket will benefit significantly more, on average.
I’m not sure that my math above is correct. I think you might only gain 1 damage from rerolling a d8 showing below average. The average should be the average of (4.5 + 4.5 + 4.5 + 4.5 + 5 + 6 + 7 + 8), which is 5.5, right?
I’m not sure that my math above is correct. I think you might only gain 1 damage from rerolling a d8 showing below average. The average should be the average of (4.5 + 4.5 + 4.5 + 4.5 + 5 + 6 + 7 + 8), which is 5.5, right?
You are right. Cutting out all the intermediate math, re-rolling all low values on a single die with "s" sides adds s/8 damage to the total, so a d4 adds 0.5 (2.5->3), a d8 adds 1 (4.5->5.5), and a d12 adds 1.5 (6.5->8). Since you take the average from (s+1)/2=(4s+4)/8 to (s+1)/2+s/8 = (4s+4+s)/8 = (5s+4)/8, in relative terms you multiply the average by (5s+4)/(4s+4), which is a number that starts at 1.2 for a d4 and rises towards 1.25 as sides increase to infinity. For 1d12, rounding the multiplier gets you 1.23.
As you correctly stated, the math is more involved for multiple dice - on 2d6, you'll always re-roll the lower of the 2. On 1d4+1d12, you should re-roll the die that's the farthest below its respective average, meaning 1 on the 1d4 being re-rolled is as valuable as re-rolling a 5 on the 1d12, so if the d12 shows 1-4, it should be re-rolled.
So I think I did some math that indicates that rerolling any die below average on 2d8+3d6 makes the feat worth about 1 damage.
It would be worth more than that, but since I don't have the formula handy, I'll just brute force it. Algorithm: Re-roll the die which is the most below its own average - if no die is below its own average, do not re-roll a die.
Baseline average: 9+10.5 = 19.50
Average with Piercer: 21.78 (rounding)
Difference: 2.28
Remember, the more dice you roll, the more valuable Piercer is. I can provide the whole formula as OP asked for, but it'll involve summing over every possible roll (using choose to generate similar rolls).
Ah, well that is more complicated than the way that I did the average: my reroll rule was simply to reroll anytime there is a roll under the average (you are adding 1 75% of the time, and less than 1 part of the remaining time). I did not account for only re-rolling the lowest die.
Ah, well that is more complicated than the way that I did the average: my reroll rule was simply to reroll anytime there is a roll under the average (you are adding 1 75% of the time, and less than 1 part of the remaining time). I did not account for only re-rolling the lowest die.
Yea that’s why I asked for help. I knew it was better than I could grasp and I lack the ability to figure out exactly why and by how much.
I was thinking about it in a way similar how the empowered spell metamagic works, but I also barely have a grasp on how effective that can be. That metamagic has been out for ages and it’s still difficult to understand.
The feat states:
PIERCER
You have achieved a penetrating precision in com- bat, granting you the following benefits:
• Increase your Strength or Dexterity by 1, to a maximum of 20.
• Once per turn, when you hit a creature with an attack that deals piercing damage, you can reroll one of the attack's damage dice, and you must use the new roll.
• When you score a critical hit that deals piercing damage to a creature, you can roll one additional damage die when determining the extra piercing damage the target takes.
some observations I’ve made is that while the 2nd
and 3rdbullet requirea an attack roll that deals piercing damage to be used, the wording doesn’t seem to require the the rerollor additional critdie to be piercing damage. It seems to be worded in a way that allows any damage die to be rerolledor chosen for a critso long as a portion of the damage is piercing damage.this got me thinking about smites, sneak attacks, and the weapon cantrips.
so my question is, can anyone explain how the single reroll mechanic might be quantified when more die are introduced? I don’t know if my guess is wrong or not but I suspect that when you add more die to a roll, and pick the lowest roll to reroll, that it might have more of an impact on the DPR than im visualizing.
for example, a level 5 high elf rogue who has chosen the booming blade cantrip would potentially deal 2d8+3d6+mod when casting that cantrip with with a rapier and sneak attack conditions met. How much damage increase is actually gained from cherry-picking the lowest roll to reroll between the 5 dice? or 11 dice on a crit…
I think the third bullet requires the extra die to be from the piercing damage for sure: “one additional damage die when determining the extra piercing damage”.
As for the reroll, well, I don’t know the stats on it but I’d imagine that the more dice you roll, the more likely you are to select a 1 or 2 to reroll, but overall you aren’t likely to add much more than half a die’s worth of damage no matter what.
Nice catch, thank you.
If you only ever reroll a die that shows below average, but then roll average on the reroll, then you’d add about 1.5 for a d6, about 2 for a d8, and about 3 for a d12, I think. From there you need to consider that you have more chances to roll a die under the average if you have more dice.
Thank you. Do you know if there’s a way to quantify the likelihood of rolling “low” on at least 1 die given that the die can be of differing types and quantities?
You know basic Probability calculations, right? Getting a 1 on a d6 is 1 in 6, or 16.6 →%. Getting a low die roll is 50% if you consider low to mean less than a die's average. Getting a 1 on each die of a 2d6 roll is 1/36. Getting a 1 on either of a d6 or d8 is 4/24+3/24 = 7/24 = about 29%. So on and so forth.
If that's all you want, the probability of a single die rolling low is 50%, regardless of number of sides. The probability of at least one die being low in "d" dice is 1-(1/2)^d.
Substantially more complicated is when you care about how low the low roll is. That's where number of sides becomes relevant; a d12 is much more likely to roll much less than its own average than a d4 is. If you want to experience this in practice, do some test rolls applying Piercer to a dagger and a musket. The musket will benefit significantly more, on average.
I’m not sure that my math above is correct. I think you might only gain 1 damage from rerolling a d8 showing below average. The average should be the average of (4.5 + 4.5 + 4.5 + 4.5 + 5 + 6 + 7 + 8), which is 5.5, right?
You are right. Cutting out all the intermediate math, re-rolling all low values on a single die with "s" sides adds s/8 damage to the total, so a d4 adds 0.5 (2.5->3), a d8 adds 1 (4.5->5.5), and a d12 adds 1.5 (6.5->8). Since you take the average from (s+1)/2=(4s+4)/8 to (s+1)/2+s/8 = (4s+4+s)/8 = (5s+4)/8, in relative terms you multiply the average by (5s+4)/(4s+4), which is a number that starts at 1.2 for a d4 and rises towards 1.25 as sides increase to infinity. For 1d12, rounding the multiplier gets you 1.23.
As you correctly stated, the math is more involved for multiple dice - on 2d6, you'll always re-roll the lower of the 2. On 1d4+1d12, you should re-roll the die that's the farthest below its respective average, meaning 1 on the 1d4 being re-rolled is as valuable as re-rolling a 5 on the 1d12, so if the d12 shows 1-4, it should be re-rolled.
So I think I did some math that indicates that rerolling any die below average on 2d8+3d6 makes the feat worth about 1 damage.
It would be worth more than that, but since I don't have the formula handy, I'll just brute force it. Algorithm: Re-roll the die which is the most below its own average - if no die is below its own average, do not re-roll a die.
Baseline average: 9+10.5 = 19.50
Average with Piercer: 21.78 (rounding)
Difference: 2.28
Remember, the more dice you roll, the more valuable Piercer is. I can provide the whole formula as OP asked for, but it'll involve summing over every possible roll (using choose to generate similar rolls).
Ah, well that is more complicated than the way that I did the average: my reroll rule was simply to reroll anytime there is a roll under the average (you are adding 1 75% of the time, and less than 1 part of the remaining time). I did not account for only re-rolling the lowest die.
Yea that’s why I asked for help. I knew it was better than I could grasp and I lack the ability to figure out exactly why and by how much.
I was thinking about it in a way similar how the empowered spell metamagic works, but I also barely have a grasp on how effective that can be. That metamagic has been out for ages and it’s still difficult to understand.