1. Introduction
2. Gold Efficient Pure Damage
3. Gold Efficient Overall DPS
4. Flat Damage (Bursty)
5. Overall BEST DPS
6. Item Scores (Based on Stat Weights)
7. Stat Weights (Based on Maths)
8. The Maths (Long and Boring)
9. Summary

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// This is more of a collection of theorycrafted builds than it is a guide. Expect to learn the BEST builds for the champion with math to prove it. Don't expect to learn how to play the champion. Eventually I will add in a guide, probably in video form, explaining how the champion is played with a few tips. Until then enjoy knowing the 100% best builds for the champion proven by maths. // SPECIAL THANKS: Credits to jhoijhoi for the template, which you can find here.

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This build is best used if you want to get the most bang for your buck and do a lot of damage in a short period of time. Its strengths lie in Mid Game and it reaches full build rather fast. If you don't think the game is going to last extremely long and you want to do more burst damage, this is the build for you.

// Increased movespeed, and attack speed. Makes it easier to dodge things in lane and also increases the damage of your ultimate, The Culling, as well as your Auto Attack DPS. These are the only boots that will actually give you more flat damage, so its the clearcut choice for this build.

// This can be a bit risky to get early game because it doesn't give you any damage, but with it you can spam abilities to your hearts content and it upgrades later on into the almighty Muramana which is the best DMG per gold item you can get. So get it semi-early and once it gets full stacks upgrade it. You can also upgrade it earlier if you want to stack it faster, but its not actually worth the gold until full stacks. And this build is all about being worth the gold.

// Massive spike in damage with this item, if you have enough gold on your first or second time back to base go ahead and get the B. F. Sword. This is the most cost effective item early game, so rush it if you can.

// This item's stats aren't actually all that great, but because its so cheap at 2300 gold its definately worth it once mid game comes around and people have a decent amount of armor.

// This is the most cost effective item for pure damage there is. So obviously we are getting it in this build. Make sure to use the toggle when you have mana to spare, which thanks to this item will be most of the time. Without the toggle this item really isn't worth it but with it its the best. So USE IT!

// The only items more cost effective than Bloodthirster we already have, so the only thing left to do is get another. Another alternative is a Blade of the Ruined King. Which technically isn't as cost effective as this item, but the activate is very helpful so in my opinion its still a good buy.

For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to.

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This build is also designed to give you the most bang for your buck, but you will have less burst damage. This builds strengths also lie in mid game but it excels in longer fights or taking out the baron/dragon. If you don't think the game is going to last extremely long and you want to do as much Overall DPS as you can, this is the build for you!

// Because Berserker's Greaves cost 100 gold less than Ionian Boots of Lucidity they actually give more DPS Per Gold, so they are the best boots for this build.

// This can be a bit risky to get early game because it doesn't give you any damage, but with it you can spam abilities to your hearts content and it upgrades later on into the almighty Muramana which is the second best DPS per gold item you can get. So get it semi-early and once it gets full stacks upgrade it. You can also upgrade it earlier if you want to stack it faster, but its not actually worth the gold until full stacks. And this build is all about being worth the gold.

// Early game this item will give you the most DPS for your gold, so rush it as soon as you can. If you can buy a B. F. Sword on your first or second time back to base get it over the Berserker's Greaves or Tear of the Goddess. Also the early sustain is a huge help.

// Now that it's mid game and people have a bit more health, this item actually trumps EVERY other item for DPS Per Gold. So it's a huge buy. Don't forget to use the Activate whenever you need to do a burst of damage or heal yourself or both! Excellent item.

// This item is the second best DPS Per Gold item after Blade of the Ruined King. Buy it once your Tear of the Goddess is stacked fully or very close to being stacked fully. The toggle is amazing on this ability so anytime you can spare the mana turn it on.

// Only 2 items can trump Bloodthirster for DPS Per Gold and you already have them, so at this point just buy another! Also this will be your third lifesteal item so your sustain will be insane.

For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to.

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This build isn't designed to give you the most bang for your buck, instead it's made to be as strong as possible regardless of the cost! No other build will outburst this one its designed to give the most damage possible in a full ability combo. If you think the game is going to last a long time and you want to do a lot of burst damage, this is the build for you!

// These are the only boots that will give you more burst damage, so they are the obvious choice for this build. BUT once you get to super late game, as this build should, you can sell these for a Zephyr as it will give you more burst damage without losing very much movement speed.

// Early game before your base AD is very high or the enemy team has a lot of HP this item will give you the most burst.

// This item will give you more overall burst than any other item in the mid game, as long as you are utilizing its passive, but as Lucian you should already be utilized your built in passive Lightslinger which is very similar. So not only is this the most burst item you can get, but it also fits Lucian's playstyle quite nicely.

// By this time people will have a decent sum of health. And late game this item gives you more overall burst than ANY other item. It works along with your passive for double procs, also the activate makes for EVEN MORE burst damage. Very good item.

// Only 2 items can trump Bloodthirster for burst damage and you already have them! So now the only thing left to do is get more of these bad boys.

For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to. BUT this build is supposed to be manmode burst, so you can also opt for ANOTHER Bloodthirster and do INSANE burst damage.

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This build is designed to be as strong as possible in the latelategame and do as much overall DPS in the long run. In an extended fight you won't find ANY other build to out damage this one overall. If you expect the game to last a long time and want to do as much overall damage as POSSIBLE, this is the build for you! This build may look a bit troll to you, and you might get flamed in your games but TRUST ME. Its 100% theorycrafted and the math below PROVES this build will do the most overall damage in a longer fight.

// These boots give the most overall DPS of any boots, so its obvious why we're getting them for this build. Also the reduced cooldowns allow for more frequent poking in the laning phase, and more dashing around in teamfights. Super lategame however you're gonna want to sell these for a Zephyr because it will give you more over DPS without losing much movement speed OR cooldown reduction.

// Early game before your opponents have much HP this item gives you the most overall DPS of any item.

// Yes I know the unique passive doesn't stack but you don't need it to. 70 AD and 25% crit brings more Overall DPS to the table than ANY other item besides Blade of the Ruined King.

// By now your enemies should have some HP, my calculations below are for a 2k HP target which by level 18 almost every champion has as a BASE health. but even before 2k this item trumps EVERY other item for overall DPS. But this item will only get better the longer the game lasts and the more HP your enemies gain.

// Being the second best item for overall DPS even with the unique passive that doesn't stack its obvious why we're getting a third. With 75% crit and a **** load of AD you're going to do A LOT of damage. Not as much burst as the previous build but overall more DPS.

For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to. But this build is supposed to be for the most Overall DPS possible and if you really want the most POSSIBLE go with yet ANOTHER Infinity Edge

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*IMPORTANT NOTE*
As far as the crit items are concerned the crit multiplier I used was 250%, so these are the numbers for if you already own an infinity edge and don't have any other increases to crit damage from Masteries/Runes. With that being said you can see even a SECOND Infinity Edge is much better than a Phantom Dancer. Also if you use the Mastery for increased Crit Damage, Infinity Edge > Bloodthirster for DPS, but not for DMG.

// Numbers are assuming the target has 2k HP and activate is used on CD. DMG Per Gold - 0.348743438671875 DPS Per Gold - 0.0709072962984375 DMG - 1415.97900375 DPS - 231.903348155

DMG Per Gold - 0.3416832531876138 DPS Per Gold - 0.0422696051712568 DMG - 1313.088742 DPS - 162.44209275

Even without an Infinity Edge this item still does more for overall damage to your abilities than Bloodthirster, its just very expensive.

// These scores are for a Bloodthirster with max stacks. DMG Per Gold - 0.3699054546875 DPS Per Gold - 0.05334379659375 DMG - 1183.697455 DPS - 170.7001491

// These numbers are for full mana, so the actual dmg/dps increase will be slightly lower than this. DMG Per Gold - 0.5264911857890476 DPS Per Gold - 0.0647267939443524 DMG - 1105.631490157 DPS - 135.92626728314

DMG Per Gold - 0.2606465903947368 DPS Per Gold - 0.0439126228276316 DMG - 990.4570435 DPS - 166.867966745

DMG Per Gold - 0.2137364609649123 DPS Per Gold - 0.0390969432105263 DMG - 609.14891375 DPS - 111.42628815

DMG Per Gold - 0.1812382642857143 DPS Per Gold - 0.0395872137321429 DMG - 507.46714 DPS - 110.84419845

// Armor Penetration hasn't been factored into this item's score. DMG Per Gold - 0.348743438671875 DPS Per Gold - 0.0296869824521739 DMG - 473.478982 DPS - 68.28005964

When comparing this item to other items, look at how much more damage they do and factor in armor reducing that damage. get rid of 35% of that armor and then factor armor into this items damage. You'll find that this item comes ahead of any item at extreme levels of armor. But sadly only at EXTREME levels. Lucian's AD ratios are just so amazing that this item is lackluster in most late game circumstances.

// The Passive has not been taken into consideration for these numbers, I leave judging the passive's worth up to you. DMG Per Gold - 0.15202988 DPS Per Gold - 0.0324379603 DMG - 380.0747 DPS - 81.09490078

DMG Per Gold - 0.1392109111111111 DPS Per Gold - 0.0239958949333333 DMG - 125.28982 DPS - 21.59630544

DMG Per Gold - 0 DPS Per Gold - 0.0221407309125 DMG - 0 DPS - 22.1407309125

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*Please Note*
The numbers for Crit Chance are using 250% crit damage, so crit will only be as good as listed below if you already own an infinity edge.

1 Attack Damage nets you...
11.83697455 more damage and 1.707001491 more DPS.

1% Attack Speed nets you...
6.264491 more damage and 1.079815272 more DPS.

1% Crit Chance nets you...
6.474753 more damage and 1.895114495 more DPS.

1% Cooldown Reduction nets you...
0 more damage and 1.4760487275 more DPS.

As far as pure damage is concerned.
AD>Crit>AS>CDR
As far as overall damage per second is concerned.
Crit>AD>CDR>AS

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NOTE - If you do read this section and find a mistake please let me know and I will look into it. THANKS!

Alright, first lets look at Lucian's base stats at level 18. (All math assumes level 18 with all skills maxed and all passive procs are triggered)
AD - 97
AS - .994357
Crit - 0

Now lets do the math to find out how much damage and DPS Lucian does with his abilities and auto attacks with no items, runes, and masteries. (No bonus AD, which is what every ability scales with except the passive)

Auto Attacks
DMG - 97 DPS - 97*.994357=96.452629

Passive - Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+72.75=312.75 DPS - 312.75/5=62.55

W - Ardent Blaze
DMG - 220+72.75=292.75 DPS - 292.75/10=29.275

E - Relentless Pursuit
DMG - 72.75 DPS - 7.275

R - The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG - (60*17.9407485)+72.75=1149.19491 DPS - 1149.19491/50=22.9838982

Total DMG - 97+312.75+292.75+7.275+1149.19491=1858.96991
Total DPS - 96.452629+62.55+29.275+7.275+22.9838982=219.5365272

Next up CDR, I'll be using 20% for easier math then later finding out what 1% yields.

Auto Attacks
DMG - 97 DPS - 97*.994357=96.452629

Passive - Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+72.75=312.75 DPS - 312.75/4=78.1875

W - Ardent Blaze
DMG - 220+72.75=292.75 DPS - 292.75/8=36.59375

E - Relentless Pursuit
DMG - 72.75 DPS - 72.75/8=9.09375

R - The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG - (60*17.9407485)+72.75=1149.19491 DPS - 1149.19491/40=28.72987275

Total DPS - 96.452629+78.1875+36.59375+9.09375+28.72987275=249.05750175

DPS Difference - 249.05750175-219.5365272=29.52097455

Now lets find out the DPS difference for 1% CDR.
29.52097455/20=1.4760487275

Now lets add 100 AD and do it all over again to see how much DPS 100 AD will bring us. (100 for easy math later when comparing stats)

New stats with 100 bonus AD
AD - 197
AS - .994357
Crit - 0

Auto Attacks
DMG - 197 DPS - 197+.994357=195.888329

Passive - Lightslinger
Adds 147.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+120+147.75=507.75 DPS - 507.75/5=101.55

W - Ardent Blaze
DMG - 220+60+147.75=427.75 DPS - 427.75/10=42.775

E - Relentless Pursuit
DMG - 147.75 DPS - 147.75/10=14.775

R - The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG - ((60+30)*17.9407485)+147.75=1762.417365 DPS - 1762.417365/50=35.2483473

Total DMG - 197+507.75+427.75+147.75+1762.417365=3042.667365
Total DPS - 195.888329+101.55+42.775+14.775+35.2483473=390.2366763

Now lets see how much more DMG/DPS that is exactly.

DMG - 3042.667365-1858.96991=1183.697455 DPS - 390.2366763-219.5365272=170.7001491

Now lets divide that by 100 to find a DMG/DPS increase for 1 AD instead of 100.

DMG - 1183.697455/100=11.83697455 DPS - 170.7001491/100=1.707001491

Now lets do AS, just like before I'll be using 100% AS.

New stats with 100% bonus AS.
AD - 97
AS - 1.988714

Auto Attacks
DMG - 97 DPS - 97*1.988714=192.905258

Passive - Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+72.75=312.75 DPS - 312.75/5=62.55

W - Ardent Blaze
DMG - 220+72.75=292.75 DPS - 292.75/10=29.275

E - Relentless Pursuit
DMG - 72.75 DPS - 7.275

R - The Culling (Shots fired is 7.5+(10.5*1.988714)=28.381497
DMG - (60*28.381497)+72.75=1775.63982 DPS - 1775.63982/50=35.5127964

Total DMG - 97+312.75+292.75+7.275+1775.63982=2485.41482
Total DPS - 192.905258+62.55+29.275+7.275+35.5127964=327.5180544

Now lets see how much more DMG/DPS that is exactly.
DMG - 2485.41482-1858.96991=626.4491 DPS - 327.5180544-219.5365272=107.9815272

Now lets divide that by 100 to find a DMG/DPS increase for 1 AS instead of 100.
DMG - 626.4491/100=6.264491 DPS - 107.9815272/100=1.079815272

Our final stat will be crit, to make the math really easy on me I'll be using 100% crit.
(For this I'll be using a crit multiplier of 250%, so these numbers for crit will be WITH infinity edge)
New stats with 100% bonus crit
AD - 97
AS - .994357
Crit - 100

Auto Attacks
DMG - 242.5 DPS - 242.5*.994357=241.1315725

Passive - Lightslinger
Adds 181.875 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+181.875=421.875 DPS - 421.875/5=84.375

W - Ardent Blaze
DMG - 220+181.875=401.875 DPS - 401.875/10=40.1875

E - Relentless Pursuit
DMG - 181.875 DPS - 181.875/10=18.1875

R - The Culling
DMG - (60*17.9407485)+181.875=1258.32021 DPS - 1258.32021/50=25.1664042

Total DMG - 242.5+421.875+401.875+181.875+1258.32021=2506.44521
Total DPS - 241.1315725+84.375+40.1875+18.1875+25.1664042=409.0479767

Now lets see how much more DMG/DPS that is exactly.
DMG - 2506.44521-1858.96991=647.4753 DPS - 409.0479767-219.5365272=189.5114495

Now lets divide that by 100 to find a DMG/DPS increase for 1 Crit instead of 100.
DMG - 647.4753/100=6.474753 DPS - 189.5114495/100=1.895114495

This next section is concerning the Blade of the Ruined King's passive. For numbers I'm going to assume the target has 2k HP then find out how much overall DMG and DPS the passive alone will give you, so I can use it to more properly find its value.

Auto Attacks
DMG - 97+100=197 DPS - 197*.994357=195.888329

Passive - Lightslinger
Adds 172.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+172.75=415.75 DPS - 83.15

W - Ardent Blaze
DMG - 220+172.75=393.75 DPS - 393.75/10=39.275

E - Relentless Pursuit
DMG - 172.75 DPS - 172.75/10=17.275

R - The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG - (60*17.9407485)+172.75=1249.19491 DPS - 1249.19491/50=24.9838982

Blade of the Ruined King ACTIVATE
DMG - 300 DPS - 300/60=5

Total Damage - 2728.44491
Total DPS - 365.5722272

BONUS Damage - 2728.44491-1858.96991=869.475
BONUS DPS - 365.5722272-219.5365272=146.0357

Now we'll test Muramana's Active toggle with full mana equal to 1927.

Auto Attacks
DMG - 97 DPS - 97*.994357=96.452629

Passive - Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+72.75+115.62=428.37 DPS - 428.37/5=85.674

W - Ardent Blaze
DMG - 220+72.75+115.62=408.37 DPS - 408.37/10=40.837

E - Relentless Pursuit
DMG - 72.75 DPS - 72.75/10=7.275

R - The Culling
DMG - (60*17.9407485)+72.75+115.62=1264.81491 DPS - 1264.81491/50=25.2962982

Total DMG - 2271.66491
Total DPS - 255.5349272

BONUS DMG - 412.695
BONUS DPS - 35.9984

Lastly I'll be testing Trinity Force's AP and passive. I'm grouping the ability power with Trinity Force because I don't plan on making an AP so it's not very important I'm just being thorough figuring out just how good Trinity Force is.

Auto Attacks
DMG - 97 DPS - 97*.994357=96.452629

Passive - Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.

Q - Piercing Light
DMG - 240+72.75+145.5=438.25 DPS - 438.25/5=87.65

W - Ardent Blaze
DMG - 220+72.75+27+145.5=465.25 DPS - 465.25/10=46.525

E - Relentless Pursuit
DMG - 72.75+145.5=218.25 DPS - 218.25/10=21.825

R - The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG - (63*17.9407485)+72.75+145.5=1348.5171555 DPS - 1348.5171555/50=26.97034311

Total DMG - 2564.2671555
Total DPS - 279.42297211

Bonus DMG - 705.2972455
Bonus DPS - 59.88644491

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Thanks for checking out my builds, before voting or commenting please at least try them or check out the math that indisputably PROVES these builds to be the best in their own category. Some of the results were actually very surprising to me, I was expecting crit to be very strong, but it turns out Lucian's crazy AD ratios (Mainly dat Passive 75% on EVERY ability) just overpowers everything else. Theres been a lot of speculation that the classic ADC builds should also be used on Lucian and although they are good and will WORK the builds in this guide are 100% guaranteed to be the best possible. According to math at least... // SPECIAL THANKS: Credits to jhoijhoi for the template, which you can find here. Top of page!