Welcome to the first post of a blog series in which I combine mathematical analysis with logic to investigate building ADCs. Why ADCs? I feel they are the easiest role to mathematically analyse and more importantly, the most relevant ones to do so. A very simplistic view of the ADC is someone who outputs consistent high DPS on the closest target. There are varying degrees of auto-attack reliance between ADCs, but many can have their damage output easily modeled.

Table of Contents

Closing Words


As a brief disclaimer, numbers alone do not give an accurate picture of what happens in the Summoner's Rift, other effects must be taken into account, those are discussed in my conclusion.

I started investigating Last Whisper after an article on Reign of Gaming extolled the virtues of building it sooner than many appear to do without offering any sort of evidence to substantiate the claims. This post investigates building Ashe. Is it more beneficial to build a Last Whisper before or after upgrading Vampiric Scepter to Blade of the Ruined King. Also included is a brief look at Bloodthirster compared to Blade of the Ruined King.

The Ashe being modeled today has the following Rune and Mastery set up.


Fleet Footwork
Phase Rush
Why aren't you taking lifesteal quints, Ashe? Frenzy is not included in the calculations but I do not believe it would affect the conclusions made. Several other simplifications have been made. The enemy's armor is constant (no use of abilities etc to increase it), the enemy does not regen life, be it from HP/5 or lifesteal/spell vamp and the protagonist is free to continue auto-attacking without having to move. A look at the process used in the model is below if you wish to verify that I am calculating everything correctly, although I have not actually put in the numbers.

Mathsy Stuff



Ashe has part of her build ( ). She can upgrade the Vampiric Scepter to a Blade of the Ruined King or go straight to Last Whisper. The first model pits her against another Ashe, she queued up in Blind Pick. Enemy Ashe has armor seals, Durability , Hardiness and Veteran's Scars . She has 1700 HP, 80 Armor.

Modeled at level 15, the results for building BotRK are as follows. It takes 7 attacks with an average DPS of 469 to kill enemy Ashe. This takes 4.94 seconds. Had she built LW, it would have taken 6 attacks, 4.97 seconds, with a mean DPS 477. Wait, how does it have higher DPS yet take longer? That is explained in the maths, but essentially it is because the killing blow has an element of overkill. The numerical difference is utterly insignificant. This is also found to be the case at level 18, where armor and health are both higher. BotRK: 8 attacks, 4.31 seconds, 474 DPS. LW: 7 attacks, 4.40 seconds, 501 DPS.

So what does this mean? Well, it is pretty irrelevant. You might duel the enemy ADC at some point, but in a team fight you will most likely someone tanky more often than the back line, someone with very different values of HP and armor. So let's look at that.

The enemy is Sejuani, her relevant masteries are Hardiness , Durability , Veteran's Scars , Unyielding , Block , Defender , Juggernaut , Reinforced Armor and Honor Guard . Her passive, Frost Armor is maintained for 20 armor at level 14 (she's one level behind Ashe) and she has armor seals. Her relevant items ( ), together with all the above give her 3207 HP and 180 Armor.

With BotRK, killing Sejuani takes 19 attacks, 10.70 seconds, with a mean DPS of 290. Had she bought LW, it would take 17 attacks, 11.23 seconds, her mean DPS is 293. The differences are again absolutely minuscule. Let's look at level 18, noting that Frost Armor now grants 25 armor. BotRK: 23 attacks, 12.40 seconds, 293 mean DPS. LW: 19 attacks, 11.93 seconds, 307 mean DPS. The differences are a bit more noticeable now, but all the other nuances of the fight could pull BotRK ahead of LW.

Hopefully, by the time you get to 18 you are close to completing your 4th non-boots item, assuming circumstance have not made an earlier defensive item more favorable it will be whatever you did not get earlier. As such, you should soon have the best of both, but there may well be a period where you lack the 4th item at high levels, depending on just how much of your experience gain is accompanied by gold.

Before I evaluate these results and open up the conclusions to discussion I would like to take a look at another scenario. Nautilus is here and he has a Randuin's Omen. His relevant items are these: . His masteries and runes are identical to those of Sejuani. The Omen is sometimes regarded as a counter to BotRK, so let's see what happens.

At level 15, we see the following. BotRK: 20 attacks, 13.24 seconds, 241 DPS LW: 17 attacks, 13.21 seconds, 246 DPS. BotRK is weaker, but it is pretty insignificant. The lifesteal and kiting potential might make it the superior choice. But what about at level 18, assuming his build is the same (additional armor would favor LW even more). BotRK: 23 attacks, 15.2 seconds, 236 DPS. LW: 17 attacks, 13.21 seconds, 238 DPS. Now this difference is pretty noticeable. The attack speed brought by BotRK is devalued, weakening it quite significantly. However, if you are attacking anything other than the person who holds the Randuin's Omen, you'll probably find that things are more even, as we saw with Sejuani.


The scenarios modeled here are just two of many, but they seem to suggest that, with the exception of attacking someone with a late-game Randuin's Omen, there is not much of a damage difference between building BotRK or LW before the other. However, there are a lot of things to consider.

Firstly, we have BotRK's superior lifesteal (we have a Vampiric Scepter if we choose not to upgrade it before building LW). This enables use to remain in the fight longer before fleeing or dying. If we spend longer in the fight we'll deal more damage, so this can make upgrading to BotRK the superior choice. Furthermore, the unique active provides further healing along with potent kiting potential, plus a burst of damage. It's the stronger item for dueling and could potentially let us stay in team fights longer.

There are a few points in favor of the LW even in the situations we modeled. Firstly, I calculated DPS for a stationary ADC. In reality, we will have to move to kite, chase or simply re-position. Mixing movement and auto attacks is a vital skill, however as long as we are moving we are less likely to be getting all of the possible basic attacks, this becomes more so the higher our attack speed becomes. As such, this devalues the attack speed from BotRK. Instead, LW edges ahead.

When you are not able to attack as frequently hitting harder becomes more important, which LW accomplishes. Notice that in the results LW always required fewer auto-attacks to kill the enemy. If you move enough to render BotRK's attack speed irrelevant or lack the skill to maintain auto-attacks while moving, LW grants better damaging potential.

Another point for LW, or rather, against BotRK, is that incoming damage from other allies will affect how much you get from the Blade's passive. If your Syndra catches an enemy squishy and blows them up with her burst, later coming to help chunk down the frontline, then you'll probably get the most from BotRK as you'll be damaging the bulky champions mostly when they are at high health and the damage from your APC will reduce the length of time you spend attacking a low health tank. However, if instead she decides that she cannot catch out the enemy ADC or APC and her burst would be best spent destroying the enemy frontline to get safely to the rear, then you may find that the tanks quickly have their health bursted below 50%, rendering BotRK weaker.

Here I have looked at Ashe, someone who gets most of her damage from auto-attacking later on, however if this were someone who makes more frequent use of physical damage abilities you would again find that LW pulls ahead of BotRK.

So, what should our protagonist have built first? The answer is that it really depends on the game. Does it look like she needs the lifesteal sooner? BotRK. Lack of threat from enemy bruisers and the like meaning that she has to move relatively little? BotRK again. Enemy tank has built an early Warmog's? BotRK. However, if the enemy has built a lot of armor early on (more than in the scenarios modeled), LW may be the better choice. Need for extra kiting power may result in BotRK's active being required, however frequent movement also reduces the damage gained from it.

Ultimately, I think that my model over-values BotRK's damage. We'll probably be moving enough to not get every possible auto-attack from BotRK's attack speed. Regardless, the differences in damage between the items are greatest later on, especially if someone build's a Randuin's Omen or Frozen Heart. If either of those items are built early, LW will definitely pull it's weight while you build up gold to build a BotRK after. However, if I am ahead and expect to be able to get my 5th complete item early, BotRK may capitalize on the advantage, in the understanding I'll get LW before the gap between items becomes significant.

When it comes to making the optimal choice, these are the factors to consider:
  • How much health do they have?
  • Do they have attack speed slows?
  • Are they stacking armor, and I mean really stacking it?
  • Do I need to stay in fights longer?
  • Am I having to move around a lot?
  • Do I need dueling power?
  • Who are my allies focusing? Do I have assassins who want to get to the back line or people who might spend more time attacking the front?

Really, the difference is not so great as to make games be won or lost on this choice, however I would err towards building Last Whisper before Blade of the Ruined King. Even if it is not much stronger than BotRK when I buy it, it might be a lot stronger before I get chance to finish the BotRK.

Bonus: BT Results

I mentioned I did a brief comparison of BT to BotRK on Ashe. Whacking the lv15 enemy Ashe, results are as you would expect, BT is stronger. 6 attacks, 3.97 seconds with a DPS of 491. This compared to 7 attacks, 4.95 seconds ands 469 DPS with the BotRK. Against Sejuani, it gives 18 attacks to kill in 11.90 seconds, DPS of 277. This is a little weaker than BotRK. BT is a fair bit weaker than BotRK against the Randuin's Omen Nautilus modeled at the end. 21 attacks, 16.32 seconds and 223 DPS. BotRK: 23 attacks, 15.2 seconds, 236 DPS.

Bonus: BT Conclusions

BotRK offers a stronger late game than BT most of the time, however low health teams may be more vulnerable to BT. If Ashe was Ezreal ( Mystic Shot spam) she might benefit from the BT more, but Volley is used primarily for applying Frost Shot later on and constitutes too small a quantity of her damage to make BT most effective. She also does not benefit as much from the snowball potential of an early BT.

There certainly are situations in which BT is stronger than BotRK for Ashe, however I would wager that they are relatively few. BotRK is going to be the better choice most frequently and is rarely much poorer. That said, recalling back to my earlier statements, devaluing of Attack Speed may push BT a little closer to BotRK, or even ahead.

Closing Words

Well, that's over 10 hours of my life gone. Whoosh, it went fast. And now I must do more maths as I return to exam revision. Huzzah... Next up is a look at how Vayne's Silver Bolts affect the value of Last Whisper, along with some other calculations for her. Anyway, I'm out for now. As always, I welcome all comments and hope to see your thoughts on this. Unless your thoughts are "Kinen's a ****bag".