### Kongai: Card Get (5/9/08)

At first it seemed like we weren't going to have a challenge after all this week thanks to some dirty pool. Fortunately, I've been using the Kongregate Sidebar Firefox Extension for a little while now (Short review: Works like a dream although I need to grab the latest version. It has no auto-update features yet.) and swiftly noticed that there was, in fact, a card up for grabs this week.

But thanks to temporal fluctuations or a prank gone horribly, horribly wrong the challenge for this week is for the appetizing Sharpening Stone. It's another green item so, yes, it's like they're trying to drive me insane at this point. I now have every green item (Six of them, in fact, since I have two Breastplates thanks to April Fool's Day.) and not a single character to use them with.

You might think I'd be frustrated by this but I'm really not. I play random decks so the cards I have or don't have don't matter to me in the slightest. And I'm probably going to continue to play random decks (At least, I will when I can find a game. There's tumbleweeds blowing through the game right now.) until release or at least until I have a lot more cards than I do now. Doesn't mean I'm not concerned about the schedule of card releases just that I'm not personally inconvenienced by it at all.

But while we get yet another green card than can only be used with green characters it's at least a good one. In order to obtain it, though, you'll need to play through the first 14 levels of Chronotron. Which is a game a lot of people like but that I'm personally indifferent towards. It's a little bit buggy with sloppy controls for my tastes - it only came out a short while ago, unless I'm mistaken, and I just don't get why newly released games that might benefit from a bit more polishing get all the attention and traffic from these badge/challenges when they could be used to spike up interest in older, more established games.

Chronotron, though, is a cutsey platformer with a lot of charm. You play as a little robot who needs to work through each level to find a missing part to your vehicle so you can move onward. The big trick is that you can travel backwards in time - most of the stages involve multiple play-throughs and co-operating with previous iterations of yourself as you work to hold switches and open doors in the right order and then get back to the start. It's a pretty neat little gimmick but, as I said, the controls get a bit maddening and the behavior of earlier robots can get a bit weird and/or laggy. Far from perfect but I can see the appeal.

Be aware, though, that it's a significant increase in difficult (For me, at least, your experience may, as always, vary.) over the last few challenges. Which is something I actually like to see. But I was actually worried that, for the first time ever, I wouldn't have the card in hand by the time I finished writing one of these things. I got stuck on level 10 - the one with four switches and a bomb. I knew exactly what I had to do but I didn't have enough time. The window was so short and the jumping so frustrating that I nearly called it quits right then and there. The only reason I made it through was by bugging my way through the collison detection. Fortunately, if you get stuck, there's a comprehensive list of walkthroughs over at the developer's.

You'll want to make use of them - if need be - because while the Sharpening Stone might not be the best item around, it's definitely a solid one. I mark the Stone down a few points Stone because, at this point, for all the Amazons I'd slot another item first (Valkyrie's for Ashi and Andie, General's for Anex, Stoneheel for Helene, Candle for Pheobe, for example although that list is by no means definitive and certainly subject to change.) but it's a strong contender for second choice. What it does is very simple: it adds 20% to your base chance to critical hit. Wearing it, you will crit 1/5th of the time more often.

Now, I've been dreading this card coming out for a while because to explain just why that's good, it's going to take some messy, complicated math as we delve deep into the game's mechanics. Not exactly my strongpoint, as much as it might seem otherwise. But before we even get to that, just what is a critical hit anyway?

Look, crits are awesome. I can argue against them from a design standpoint but from a pure, bathing in the blood of my enemies standpoint, you can't beat bigger numbers. In Kongai, every time your character lands an attack they have a chance to do extra damage. That's a critical hit. Although there are some skills that alter the formula somewhat, when you critically hit you'll do 1.5 times your normal damage or 50% bonus damage. So, if your skill causes 20 damage normally, when it scores a critical hit - complete with screen rocking and flashy graphics and bigger numbers - you'll take 30 health off your opponent's bar. I'm seeking to confirm this through running a lot of observational trials and jotting down the results but the game's guide says that the base crit rate is 3%. I don't have nearly enough data yet to say one way or the other but it wouldn't surprise me given the numbers I've got already. So, for most attacks you will score a critical hit 3 times out of every 100 swings. On average. There's nothing stopping you from scoring a critical hit three times in a row or not landing one at all in a thousand swings, it's just odds and probability at that point.

When you score a critical hit, it seems that your opponent's resistence is taken off *after* the raw damage is calculated. So, for example, against someone with 4 resist that earlier 20 damage skill would deal only 16 damage. When you crit, though, instead of get 50% of that 16 damage for a total of 24 you would get 26. Or 1.5 times the 20 minus the 4 instead of 20 minus 4 times 1.5. This works with multi-hits, too. Say our skill did 7x3 damage instead of 20. Raw damage from a crit, then would be 33 (Rounding up, which I'm not sure the game does) or 11x3. Against a resistance of 4, that crit should be 21 (Or (11-4)x3) instead of 14 (Or ((7-4)x3)x1.5) although I've yet to really confirm this.

There are some skills for which this is different. Popo's Slingshot, for example. Which not only has a dramatically increased chance to crit but also a larger multiplier. A slingshot crit will hit for 3x instead of the normal 1.5x. But even skills like Voss's Double Slash and everything else that says "high chance for a critical hit" are different. The actual crit chance for these skills is undocumented (I'm working on it.) but the tutorial says to assume they're around 25%.

Okay, so, to review, critical hits make your attacks stronger - although they won't do it every time - and while the normal rate for this to happen is a paltry 3%, certain skills have a 25% chance (Or are over 8 times as likely to land a critical hit). What Sharpening Stone does, then, is turn any normal attack skill into a skill that reads "high chance for a critical hit." There's a difference between 3+20% and 25% but it's negligible (And, from what I can tell, yes, the bonus is additive rather than multiplicative. In other words, with a Sharpening Stone it's 23% not 0.03x0.25 or whatever. You get one chance to crit and the Stone just shifts that number higher.).

It makes those high-crit skills even better, of course, but there's only one of those in the Amazon line - Andromeda's Trueshot and good luck getting that one off during a game. So, really, it's all about making your normal attacks crit about a forth of the time. And that means your card is doing more damage.

It's not as straightforward as slotting a General's Insignia, though. Not only is it a random chance instead of a straight bonus - again, the Stone isn't doing you any good if you get on a streak of bad rolls but, then, it could be doing you a lot more good than normal if you're lucky - but you also don't get as clear a picture of how much extra damage you're causing. The Insignia gives you +6. Although, the Sharpening Stone works on both multi-hit and single-hit skills and is, therefore, better that the General's in that regard, to understand how much damage it's adding requires you to do a bit of mental arithmetic in order to figure out your ideal damage, you expected damage, and just how much the Sharpening Stone is raising that. Because when it comes to damage, it's not important what's listed on a card's backing but, instead, the amount of damage you'll actually be causing. And that means you need to think in terms of averages.

Let's start easy and say you have a skill that hits all the time and deals 30 damage. Each and every time you spend whatever energy it costs it will connect with your opponent and cause 30 damage a hit. That's your average. All the damage you do divided by all the times you do it, no matter how much or how many rounds it'll always come back to 30 (And you in the back, shut up about switching. We're assuming a training dummy here that's not going to do anything but get hit.). Simple, right? Okay, now what if instead of having 100hit it had, say, 90hit? You'd still deal 30 damage when you connected but your blows wouldn't land every single round. You'd miss a tenth of the time and, over time, that hurts your overall damage. In the long run (Thanks to the Law of Large Number) you'd average not 30 damage but, instead, about 27. Or, another way of putting that is your base 30 multiplied by that 90% rate. Your expected damage, then, is your actual damage controlled by the rate at which you can expect to hit.

Okay, now forget about missing. That lowers your damage. Instead, add in a crit chance which will increase that damage. Say our skill is weird and crits 10% of the time. Every tenth hit you don't whiff but, instead, you smack them for 45 damage or an extra 4.5 damage over that long haul. You might think this means your average damage raises to 34.5 but you'd be wrong. You'd be saying you're deailng your raw damage of 30 plus 10% of your critical damage of 45. But each critical hit replaces a normal attack so what you want to count, instead, is not the crit damage but the bonus damage that each crit adds on top of your normal damage. Which, in this case, is 15. A 30 damage skill that hits all the time with a 10% chance to crit for 150% damage would average 31.5 damage. Or the base 30 plus 10% of the 15 bonus damage.

A 3% crit rate, by the way, on our imaginary skill would work out to about 30.5. So, as you can see, the natural crit rate might swing a particular battle your way but it's not going to be making much of a difference overall. The extra damage from a 25% crit rate? About 4. Which actually might. It's not as good as the raw damage off a General's and most of the Amazons don't have the multi-hitters that would really take advantage of it, but you have to take into account the difference between the steady pressure of consistent damage versus the lethality of a burst of damage - the extra damage from a crit arrives not in packets of 4 but in large chunks that can carve into your opponent's life bar. It's the difference between signing up for an investment with a steady interest rate and one where you might double your money or you might wind up with nothing. General's lets you pound on people. The Stone lets you kill them, every so often.

It gets a little more complicated than that, though. Because not every skill in the game has a perfect 100% hit rate. You will miss with most of them sometimes and you don't score critical hits when you miss. Let's go back to our 30 damage skill and say it, again, has 90hit to go along with 10crit. Now, you still average 31.5 damage when you connect. But you don't connect all the time so your actual damage output slides downwards again. Just like before, you can take that raw damage of what you can expect to do if you hit every single time and multiply it times your hit rate in order to figure out what you should actually be doing. In this case it's that 31.5 times 90% or about 28 damage. And that 10% crit has added only a single point of damage overall (Again, it's a bit misleading to think of it this way since crits are not best over the long haul but, instead, in discrete units where they're outperforming their expected probabilities. You might add only 1 damage as your games played stretches out into infinity but if you land three crits in a row then you've added 45. And that's a lot.). If our skill hit only 75% of the time then your damage would fall even lower and so, too, would the added value of each crit (The relative value, though, might actually go higher but this is where the math gets confusing for me.). So, adding crit chances are best on skills that have a high percentage to hit. More hits equals more hits which equals more damage.

This difference between expected damage and actual damage (Along with the lack of multi-hit skills in the Amazon line) is the reason why I think it's better to stick with General's instead of slotting the Stone - you might get the occasional eye-popping number but the Insignia will be adding objectively more damage to a card like Ashi. A Helene, an Andromeda benefit more from it but I think you can make a case that they'd benefit more from other things. I haven't run all the numbers and calculated the differences on each and every skill but my gut feeling is that 20% is a little too low to be better than the Insignia (My gut also says that anything higher gets ridiculous so it's fair where it is. Again, it's the difference between rolling the dice and playing it safe. The Stone is a little more swingy and making it proc even more crits so it's equivalent to the General's makes it even more so.)

Got it? Well, don't worry if you don't. The only thing you need to know is that more damage is good and Sharpening Stone adds significantly to your your damage. How much depends on what skill you're using - skills that hit more often are going to crit more often and, therefore, get that bonus damage more frequently.

Now, go out there and get it.

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