My Halfling Hexblade Warlock 1 / Eldritch Knight Fighter 3 has come across two magic weapons:

  • a longsword of life stealing (3d6 necrotic damage extra on crit)
  • a +1 longsword.

What weapon combination works best for consistent damage per round for a full combat (3-5 rounds).

Traits that might influence it:

  • two weapon fighting (Fighting Style)
  • a familiar that can give me advantage.
  • green flame blade and eldright blast cantrips (rest are non combat).
  • my first level was in Warlock
  • my stat bonuses are: STR 0, DEX +3, CON +1, INT +2, WIS 0, CHA +2.

Everything else is standard from the class/race; things like disadvantage on heavy weapons, Hexblade's Curse and Action Surge are in play. I'm thinking of taking the Dual Wielder feat next level but that depends on the results of this question.

I could buy every other non-magic simple/martial weapon, but I have short swords at my disposal already.

  • \$\begingroup\$ Are you using the hex spell? That will make a difference in DPR calculation if you frequently use that. Also, what are your cantrips? (I am guessing Eldritch blast, but that would be nice to confirm). \$\endgroup\$ Dec 19, 2018 at 14:15
  • 2
    \$\begingroup\$ Do you want this answer for your current level (4) or for level (5) once you have the feat, or for level (6) when you have two attacks. \$\endgroup\$ Dec 19, 2018 at 14:25
  • \$\begingroup\$ @KorvinStarmast just for the current level but adding the other level on to it is up to you. \$\endgroup\$
    – darnok
    Dec 19, 2018 at 14:33

5 Answers 5


I'm not going to do a deep dive into specific mathematics here, because you're already heading down a non-optimized route. What I am going to do is try to give you a feeling for the variables involved and where you should go in the future.

Weapon Choice

Considering the racial choice locks you out of effective use of Heavy weapons, there's no debate that using two weapons is the best choice for weapon DPS. As for which weapons, or if weapons are even the right answer, that really depends on how much more Warlock you're going to take vs. how much Eldritch Knight.

For this character, magic shortswords would be better here, because you could use them with Dexterity via finesse and together via light... but you've got what you've got. The difference between 1d6 vs. 1d8 weapons averages a single point, so it's almost not worth considering if you're not super-optimizing. The difference of +1 to hit is far more significant to your damage output, though - use the +1 longsword for now.

If you're committed to Dual Wielder at some point (perhaps for the aesthetics), twin rapiers is a better option. For survivability though, you may want to stick to one and use a shield. Right now, you could use the sword two-handed via versatile for 1d10, but again that's just one more damage on average. A corpse deals no damage, so unless this character has weapons in both hands, it should have a shield in the other. (Great Swords and Great Axes in conjunction with feats have other benefits, but they're no-go for a Halfling.)


Right now, the character can effectively use one of those longswords with Charisma via Hex Warrior. It's not as good as Dexterity, but it is better than Strength. You can't use the other one simultaneously without taking Dual Wielder, because they don't have the light property.

If you go to Warlock 3, you'll be able to pick up Pact of the Blade, which does allow you to use two weapons with Charisma (one via Hex Warrior, one as your Pact Weapon). Of course, you'll need Dual Wielder for that.

Whatever you do, don't neglect Hex. An extra 1d6 of damage on every hit will add up quickly.

Eldritch Knight

If you take Eldritch Knight to 7, you'll gain...

War Magic Beginning at 7th level, when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.

That will let you take the most advantage of green flame blade (herein "GFB"). For the purposes of conversation, I'm going to look at Eldritch Knight exclusively...

  • In Tier 1, GFB adds a small amount of damage to a secondary target. The problem is that GFB uses the Cast A Spell action, which doesn't trigger the bonus action attack from using two weapons (you have to use the Attack action). If you're using two weapons, GFB actually lowers your DPR. Assuming a d6 weapons in each hand with the fighter style and two hits, but no feat, you're looking at damage of 2×(1d6+attack stat) with casting stat in damage to a secondary target.
  • In Tier 2, a fighter gains Extra Attack. While GFB does increase to an extra 1d8 to the original target and the secondary, you're also likely to have Dual Wielder by this point and be swinging d8 weapons.
    • GFB hits for 2d8+attack stat to the primary and 1d8+casting stat to the secondary.
    • TWF hits for 1d8+attack stat to up to three targets, each of which can potentially be a critical hit.
    • At L7, an EK eeks out a bit more, getting the hit from GFB, damage to a secondary target, and another basic weapon attack from War Magic. Note, however that War Magic is a swing with the same weapon. You don't even need two weapons for this, nor do you need a feat.
  • In higher tiers, the fighter gains additional attacks, but GFB gains two damage dice each level. Unless you're crit fishing (or need high rolls to hit at all), GFB is a good choice. If you're having a hard time hitting, more rolls is better (because any damage is good damage), so you wave off GFB.

Flame Tongue, Frost Brand, etc.

If you're going for big damage as a fighter, you want a weapon that deals extra damage on every hit. Your extra-on-a-crit weapon is nice, but critical hits aren't your bread-and-butter. The dip of warlock helps because of Hex, but full-time-extra-dice is something you really want.


If you're leaning more heavily towards Eldritch Knight (at least 7): Get a more appropriate weapon, and don't bother using two. Use a shield instead.

If you're leaning more heavily towards Warlock: Get a more appropriate weapon, and go ahead and use two. Consider leaning more towards Eldritch Knight.


At your current level, use the two weapon fighting (with one caveat)

You mentioned in a comment "at the current level" so we'll do simple numbers first, and then the more tricksy ones. You took the two weapon fighting style, so I make the assumption that not using a shield is your choice: you want to maximize DPR. I also assume that you cast Hex, a first level spell, and have that as a known spell. Not having Hex reduces your DPR for this character. Under that assumption ...

  • Average damage for a long sword wielded two handed: 5.5 and Hex once and Green Flame Blade(GFB) once

  • Damage for sword of life stealing (includes 3d6 on critical hit) is 5.5 + (.05 x 10.5) = 5.5 + .525 = 6.025. The +1 Long Sword hits more often, and its average damage is 6.5.

  • Average damage for two short swords: 3.5 twice (7.0) and hex twice (3.5 x 2 more)

  • Using the Hex spell, you add 1d6 to each successful hit. (Hex stays up for an hour, but it's a concentration spell).

  • Using Green Flame Blade, you add 2 on one successful hit (to another target; if there aren't two targets, there is no benefit).

    Short swords are better at the moment unless the target has resistances, or immunities, to piercing attacks. Your attack modifier is better with the short swords (Dex, +3) than the sword of life stealing; it is the same using Charisma (+2) and the +1 longsword. You want to hit more often to trigger Hex damage more often. Your DPR should be higher since you'll hit more often, and your second short sword gets the damage from your Dex bonus.

Immune/resistant targets: use the long sword +1.

If it is a werewolf (or other immune creature) go with the +1 Long sword. Hitting more often makes up for any hope for a nova, particularly in short fights.

With all of that said, if your familiar survives for all 3-5 rounds (in my experience they die fast when getting into combat to help) the sword of life stealing becomes more attractive but still comes out second best.

If the familiar can help for each round, does that change the choice?

We will use an Armor Class of 15 for the purposes of this comparison. With the attack at advantage, the unmodified target number for the first short sword attack (+2 prof, +3 dex) gives target number of 10: 79.75 chance to hit (same with +1 long sword, +2 Prof +2 Cha). For the sword of life stealing (+2 Charisma, +2 prof) the target number is 11 which is a 75% chance to hit. The second short sword attack has chance to hit (TN of 10) of 55%.

Note: GFB yields 2 per successful hit (if there is a second target close enough;) and Hex yields 3.5 per successful hit.

What does that do to DPR? The linked table gives a critical hit chance of .0975, up from .05, for attacks with advantage.

  • +1 Long Sword (adv): .7975*(5.5 + 1 + 2 + 3.5 + 2) + .0975*(5.5 + 3.5 + 2) = 12.237
  • sword of life stealing (adv): .75*(5.5 + 2 + 3.5 + 2) + .0975*(5.5 + 10.5 + 3.5 + 2) = 11.846

  • Short Sword #1: .7975*(3.5 + 3 + 3.5) + .0975*(3.5 + 3.5) = 8.657

  • Short Sword #2: .55(3.5 + +3 + 3.5) + .05 (3.5 + 3.5) = 6.2

    14.857 DPR, so the two short swords recommendation stands.

    But not by much. A problem becomes losing concentration if you get hit. That significantly reduces Hex's benefit in the fight. Make that Con save!

Your Character level 5 choices: goal is DPR boost.

You can choose either another level in Warlock, or another level in Fighter.

If you boost your Fighter Level, at character level 5, equip both of the magical long swords. The Two Weapon Fighting(style) and Dual Wielder (feat) puts two 1d8 long swords in place of two 1d6 short swords, both are magical, and you get the occasional crit boost from the sword of life stealing. The feat improves DPR, and increases your armor class by 1 - which you need since you are not using a shield in your quest for more raw damage output, and you want to keep Hex up. (I suggest the +1 long sword as your primary attack so that you trigger Green Flame blade more frequently. The first damage boost for GFB comes at character level 5. The TN for the bonus action attack is 13 (sword of life stealing) so 40% chance to hit. We'll ignore the damage to a secondary target for the moment, since your target now takes 1d8, which is 4.5 average, on a hit, but your two weapon fighting style precludes that. Hex lingers on in its effectiveness.

  • +1 Long Sword (adv): .7975*(4.5 + 1 + 2 + 3.5) + .0975*(4.5 + 3.5) = 8.7725
  • sword of life stealing (no adv, no Cha boost): .40*(4.5 + 0 + 3.5) + .05*(4.5 + 10.5 + 3.5) = 4.125

    12.8975 DPR and the +1 to armor class.

If you take the next level in Warlock, and get the Agonizing Blast Invocation, your ranged attacks (due to casting cantrips based on character level) are each 1d10 +2, and you get two of them. While I am not sure of the mix of ranged versus melee attacks that you want, consider this: at range, for the same armor class of 15, your DPR looks like (with Hex, without GFB)

Chance to hit with advantage:

  • (Agonizing) Eldritch Blast #1: .7975*(5.5 + 2 + 3.5) + .0975*(5.5 + 3.5) = 9.65

  • (Agonizing) Eldritch Blast #2: .50*(5.5 + 3.5 +2) + .05 (5.5 + 3.5) = 5.95

    Total of 15.6 expected DPR, and you can use a shield for a +2 to armor class rather than +1.

At levels six through ten

If you decide to take level 5 in Eldritch Knight, you open up the two attacks plus a bonus attack with the long sword, so your DPR becomes more favorable with the Dual Wielder Feat until level 11 when you get your third eldritch blast.

  • \$\begingroup\$ Does your calculation account for gaining advantage with a longsword wielded two-handed via the familiar the character has? \$\endgroup\$ Dec 19, 2018 at 15:19
  • \$\begingroup\$ it would apply to one attack (thats what the Help action does). The math isn't as simple as you imply \$\endgroup\$ Dec 19, 2018 at 15:22
  • \$\begingroup\$ @DavidCoffron Please check the numbers, I used the tables provided in the Q&A that discusses advantage. \$\endgroup\$ Dec 19, 2018 at 16:40
  • \$\begingroup\$ looks right to me. \$\endgroup\$ Dec 19, 2018 at 16:43


You want to boost Charisma as quickly as possible, after you've gotten the Dual Wielding feat. Dexterity is useless on Longswords, and since you can have two "Hexblade Weapons" (one being your actual Hexblade weapon, the other being a Pact of the Blade weapon) you'll want to just use Charisma for both anyways.

When given an option of which weapon to use, prefer using the Life Stealing longsword, as its overall DPR is slightly higher than the +1 Longsword in most scenarios. The overall DPR between them is very similar though, only differing by about ~0.5-1.0DPR depending on the AC of your target.

I'll be presenting stats both with the use of both of your Hex abilities (Warlock spell Hex, and Hexblade feature Hexblade's Curse), and without the use of said abilities. Generally speaking, using those features together adds 50%-100% to your overall DPR, but it depends on how you use your Action on those turns.

Also, I'm taking you at your word that your Familiar grants you Advantage to attacks, so all stats were calculated with attack rolls made with advantage; these stats will be quite different without Advantage.

Make use of Green Flame Blade, use the Life Stealer as a primary weapon and the +1 Longsword as Offhand

I'm going to try to limit the stats in this post to only the essentials. I ran the stats using a personal-use Probability Engine to calculate DPR values, and had to consider a lot of class level combos to work out the optimal build. So understand that I've excised a lot of stats simply for the sake of brevity.

The following will be your total DPR chart by [important] level milestones, in the order I'm suggesting you advance at.

DPR Chart

Some important notes:

  • Hex and Hexblade's Curse add a lot of damage. Use them when you can.
  • The order of priority on attacks should be
    • Attack Action + Bonus Action Attack below level 12 (EK8/HB4)
    • Green Flame Blade + Bonus Action Attack above level 12 or at level 5 (EK4/HB1)

Level 5 (EK4/HB1)

You take the Dual Wielding feat at this level, so your Charisma is still 14/15, meaning your modifier is +2. This benefits your Life Stealing longsword, which you'll want to make your Hexblade weapon.

HOWEVER. You don't have your Pact of the Blade feature yet, meaning your offhand Longsword is using Strength as your modifier instead of Charisma, weakening it. Bear that in mind.

Level 6 (EK5/HB1)

Here you'll gain Extra Attack from Eldritch Knight. For awhile, using your regular Attack Action is going to be superior to using Green Flame Blade (though you may still prefer it if you are up against two targets)

Level 8 (EK5/HB3)

At this level, you gain Pact of the Blade from reaching level 3 as a Warlock. This will allow you to add your Charisma Modifier to your +1 Longsword. That's important because...

Level 10 (EK6/HB4)

It doesn't matter (much) which order you gain these levels in, as either way, your charisma will bump up twice in two levels, giving you a lot of extra power to your attacks. Bear in mind that Eldritch Knights, as Fighters, get an extra ASI at level 6!

Level 12 (EK8/HB4)

Level 7 as an Eldritch Knight is very important. It's getting you an extra Bonus-Action attack after you use Green Flame Blade, and (for you, because it's character level 11), it's bumping GFB up to its third tier.

Then, level 8 as an Eldritch Knight lets you get your Charisma up to 20. At this point, because Green Flame Blade is now dominating the DPR, there's little reason to gain more Eldritch Knight levels.

Levels 13-20 (EK8/HB5-EK8/HB12)

At this point, you're just building up to getting the Lifedrinker invocation so that each of your weapon attacks can benefit from double-dipping your Charisma bonus. In the meantime, you'll want to focus on lots of utility spells from the Warlock class to help deal with immobility or grounding aerial attackers.

Lifedrinker vs Extra Attack (2)

I've omitted the specific stats for the sake of brevity. The short version is that Lifedrinker stacked with Green Flame Blade is about ~4.3DPR better than being able to attack 4 times without it. They're very close overall though (within 8% difference) so if you prefer playing up to a higher level Eldritch Knight instead, you aren't losing everything to do it.

Final Remarks

There are a LOT of combos to test with this particular arrangement, and it's possible I haven't caught a particularly strong combo between these two classes; if I missed something, point it out in the comments and I'll try to integrate it into my testing.

  • \$\begingroup\$ Rather than an image, you may want to use MathJax to format the contents as an actual table (or two - one without Hex and one with Hex) to make the answer more accessible. \$\endgroup\$
    – V2Blast
    Dec 20, 2018 at 1:27
  • 1
    \$\begingroup\$ @V2Blast I tried to do that at first, the original version of the table made the Stack Exchange interface too laggy to work with. I reduced it down to a manageable size after I'd already committed to rendering it in Excel instead. \$\endgroup\$
    – Xirema
    Dec 20, 2018 at 2:36
  • \$\begingroup\$ Ah, makes sense. \$\endgroup\$
    – V2Blast
    Dec 20, 2018 at 2:37

Ok, lotsa variables in here. let's see what we can't unwrap it into.

First off, GFB vs two-weapon:

GFB deals charisma damage on a second target. In your case, that is 2, roughly equivalent to 1d4. Even a dagger, factoring in dex, outdoes it. Single-target, GFB doesn't even contribute. So, for your current level, always favor second--weapon attack as a bonus action.

But, starting next level, its damage increases by 1d8 on both targets. That's 1d8+2 for the second target (more than any light weapon, and only surpassed by a rapier for you as far as single-handed go, not an option off-hand), and 1d8 extra on your main target. And since you could two-hand your longsword, for an average +1 damage, your main target would take 1d8+1 extra damage through GFB. That's compared to the best-case 1d6+3 of a nonmagical scimitar (or similar) for an off-hand, which is 1 less damage. That means it's not even a competition against 2 enemies, while TWF only barely wins out on a strict 1 enemy (a rare scenario).

By 11th level, GFB gets another 1d8, getting way ahead, and if you increase your charisma mod by at least 1 point, GFB breaks even against solo too, at 5th level (which you love, cause at that point you can go all-out on charisma for all the other goodies it brings to your spells and not bother with dex). Also worth noting the different damage types, so depending on what you're up against things might change. But, this is true both ways (and you should keep your alternative at hand; versatility is not overrated).

Also, if you go two-handed longsword+GFB, and have to use your bonus action for something else (like hex/curse management) you get to keep your average +1 damage.

So overall, stick to TWF for this level then either go GFB + cha for the long term, or stick it out with TWF till 11/cha gain, using your GFB when it gets 2 targets. Would certainly go GFB one-handed with a sidearm just in case.

Next, we gotta pick a longsword. Both deal 1d8 + str (or more likely cha, based on Hexblade), so we should focus on enchantments.

+1 means a fixed +1 damage per hit and an average 5% hit-chance bonus (can get more accurate if you factor in actual enemy AC, but I wouldn't bother). That's an average total of 1.4 damage bonus on the +1, plus 5% of any other bonuses (like GFB's total, 2-enemy damage). Overall, it is your 'team player' and gets better the better you get.

3d6 on a crit amounts to 0.525 damage per attack, disappointingly enough, or 1.45(ish) on advantage. It is much more satisfying when it happens (if often overkill), and doubles when you use your Hexblade's curse, but otherwise, it loses out to the +1 (if you figure that the non-certainty of familiar advantage is worth 0.05 damage, which I think it does).

So if you want raw numbers and not building to increase crit range, +1 starts stronger and only grows. Go with that. If you do build crit-range, and don't get any fancy tricks beyond GFB to increase damage (including single-weapon, charisma increase or any other bonus), you could go with necrotic.

Finally, off-hand option. Since there is no Light weapon that is not Finesse (except club, but that's crappy, and handaxe, not better than your finesse options) we're not interested in Hexing your off-hand, and can hold onto the longsword. That means 1d6, for scimitar and shortsword, is as good as it gets. If you want offhand for options, not baseline damage, go club for bludgeoning, shortsword for stabbing, and whip for reach. Draw to avoid fire resistance or exploit vulnerabilities.

So, TL;DR: Shortsword and +1 for now. +1 and GFB on your next level+cha gain, or at 11th, or versus 2 enemies at 5th+, or after just 5th level in the name of action-economy-reliability and ease over marginal damage gain. Shortsword available for base versatility. Shortsword, club and whip available for increased versatility.

All that assuming no invocations providing you with their own weapons, and I'm not sure if any of those survived past UA.

Also keep in mind: without Improved Pact Weapon, two-weapon fighting prevents you from casting focus-requiring spells.

  • 1
    \$\begingroup\$ It's difficult to parse your answer as a wall of text, but a couple quick notes: 1)Try and organize your answer so that it's more readable; 2) A +1 Shortsword was not one of OP's options; 3) OP is an Eldridge Knight, not a Champion; 4)Really great point about TWF preventing you from casting material component spells without Improved Pact Weapon. \$\endgroup\$
    – NotArch
    Dec 19, 2018 at 17:34
  • 1
    \$\begingroup\$ Oh, I really should. I meant the +1 longsword, where'd I mess up? In tl;dr, I meant a mundane shortsword and a +1 (longsword) :P Oh, missed the EK point! Going back to edit. thanks! \$\endgroup\$ Dec 20, 2018 at 9:43

Well, it depends one some things.

Either way you need to choose which weapon you bind to with your Hexblade Blessing.

The +2 damage and to hit (ability score bonus from Charisma) will make the bound weapon superior to other one-handed weapons (Though a close call compared to a shortsword).
For a single all-out fight (against a 15+ AC boss), the necrotic longsword wins versus from the +1 longsword. The bonus adds about 0.5 damage per round without advantage. However, lets not forget that the base damage per round is also reduced by 25-50% (based on enemy AC) due to missing attacks. When using the hexblade curse, your increased crit-range doubles this bonus.

When not using your hexblade curse the +1 to hit and damage is superior.
Where relevant, you could spam green-flame blade.
You can either leave your offhand empty for spellcasting and about 0.7 damage (two-handed fighting) or for a shield, depending on what spells you use and how much you need a shield's AC bonus.

Dual-wielding short swords is still also an option for you for a very slight damage increase. (Compare to Hexblade curse, GFB and Second Wind, as well as not benefitting from action surge. The increased damage only works during long fights.) This would also sacrifice the free hand for spellcasting / the shield.

I suggest using the +1 longsword normally. If you know a tough fight is going to show up soon (like if you are going to ambush an enemy leader) you can rebind to the necrotic longsword.

In regards to the dual-wielder feat: this lets you dual-wield a rapier alongside your currently bound longsword (same choice as above), but quite frankly, the damage increase from this doesn't exceed the increase from taking +2 Char, especially since GFB is going to deal a bunch more damage at level 5. After the level up beyond that, your extra attack will push the decision even further towards one longsword + free hand / shield.

In short: currently the highest damage would be twin shortsword though only in long fights (5 + rounds). That would however limit your use of spells, prevent GFB, and contest with hexblade curse and second wind.

But: After you level up, the fact that you can't make an offhand attack when using GFB(at level 5), makes the longswords (with Increased charisma) a clear winner.

Second best would be either the +1 longsword (bound with hexwarrior) for multiple fights between short rests, or the necrotic longsword (bound with hexwarrior) for a single fight per short rest. Keep in mind about the longsword: it takes a long rest to switch your binding.

Some supportive math:
Assuming AC 12, 4 rounds of combat, no advantage, no target for GFB is in reach, no use of second wind, no loss of hex midfight, no shield on longsword

Longsword+1 [hits on a roll of 6+->75%. Damage: d10(5.5)+2+1]
Round 1: Hex(+d6(3.5))+attack. ~9 damage +0.45 for crit
Round 2: hexblade curse(crit range)+attack+action surge attack. ~18 damage + 1.8 for crit
Round 3: attack: ~9 damage+0.9 for crit
Round 4: attack: ~9 damage+0.9 for crit.
Total: average damage of 49.05

Necrotic longsword [hits on a roll of 7+->70% damage d10(5.5)+2(+further 3d6(10,5) on crit)]
Round 1: Hex(+d6(3.5))+attack. ~7.7 damage+0,975 for crit
Round 2: hexblade curse(critrange)+attack+action surge attack. ~15.4 damage+3.9 for crit
Round 3: attack: ~7.7 damage+1.95 for crit
Round 4: attack: ~7.7 damage+1.95 for crit.
Total: average damage of 47.475 (lower then +1 longsword without hex)

Dual shortsword [hits on a roll of 6+->75%. Damage:d6(3.5)+3]
Round 1: Hex(+d6(3.5))+attack. ~7.5 damage+0.35 for crit
Round 2: hexblade curse(critrange)+attack+action surge attack. ~15 damage+0.7 for crit
Round 3: attack+offhand attack: ~15 damage+0.7 for crit
Round 4: attack: ~15 damage+0.7 for crit.
Total: average damage of 47.8

Final notes on math:
The necrotic long sword doesn't win from the +1 one in this math due to kinda low AC (higher AC means you miss more of the non-crits). They are about equal in a single fight at about 15 AC.
The dual wielding short swords win at 5 rounds and beyond.

  • \$\begingroup\$ Can you support this with the associated math proving it? \$\endgroup\$
    – NotArch
    Dec 19, 2018 at 14:35
  • \$\begingroup\$ Thats some very complicated math, but i'll give it a shot in a few minutes \$\endgroup\$ Dec 19, 2018 at 14:36
  • 3
    \$\begingroup\$ I know - and it's why i'm not answering - but it's pretty much a requirement otherwise it's just, like, your opinion man :P. \$\endgroup\$
    – NotArch
    Dec 19, 2018 at 14:37
  • 1
    \$\begingroup\$ @NautArch Lebowski reference? :D \$\endgroup\$ Dec 19, 2018 at 14:39
  • 1
    \$\begingroup\$ OK, I have cleaned up your formatting, and got rid of the commas (rather than full stops) that indicate decimal places since you had used them inconsistently and the convention here is to use a full stop for a decimal place. Please review this answer again; I also cleaned up some typographical errors. \$\endgroup\$ Dec 20, 2018 at 15:13

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