# Exploding Dice!

**1. INTRODUCTION:**

*Written By*

**GAMESLAYER989**

*Helloeveryonethisisme:*

**Gameslayer989**aka

**NooBrainer**and welcome to my first ever

**guest**

**article**!

One thing that’s near and dear to my heart in this game is the

**original Yoda**, the Legacies version, the guy who cost me at least

**£80**and that's totally not the reason why I love him so much. I don’t love people for the money, let’s make that clear! Even though he is absolutely fantastic at bringing home the big bucks with

**his Special chainin**g and all, but you know what? This article isn’t even about him! It’s instead about another great thing that

**Star Wars Destiny**uses:

**DICE**. Not just any dice though, oh no, I want to discuss the

**dice of infinite potential: EXPLODING DICE**!

**2. EXPLODING DICE:**

**Exploding dice**are dice where IF you roll their

**maximum**

**value**, you get to

**roll the die again**and add up the results. If that next roll also rolls its

**maximum**, you get

**another roll**, and this keeps going on

**until you stop rolling that value**. What does this have to do with

**Star Wars Destiny**? Well, we kinda have a similar thing with a couple of dice in the game, and two prominent characters with

**conditional**

**exploding**

**dice**have started to regularly feature at top tables across the globe.

I am of course, talking about

**Spark of Hopes Rey & Kylo - Bound By The Force**. These two characters both have

**Specials**which allow you to

**resolve**

**the**

**die**

*and*

**reroll**

**it**

*instead*of removing from the pool. Yes they’re conditional, but these can technically go

**Infinite**- especially if pitted against each other!

**Rey's Special**

*gives*

**2 Shields**,

**Kylo's Special**

*breaks*

**2 shields**.

**Rey's Special**

*gives*

**2 Shields**... LONGEST. FIGHT. EVER!

So this begs the question, what’s the average value of their dice? Obviously

**we can’t say the value of each symbol is equivalent**, but for the

**sake of simplicity**in our maths we shall assume that

**every Symbol is equal in value**. This means that

**Rey**and

**Kylo's**

**dice**each have a

**1/6 chance**to roll a

**Blank**,

**3/6**a

**1 value side**,

**1/6**for a

**2 value side**, and

**1/6**for a

**Special**, which is

**2**and a

**reroll**. Without the

**reroll**this would give us an average value of

**7/6**, about

**1.2 per die**.

However, that

**Special**is obviously

**worth more than a 2**. It’s worth

**2 + the average value of the die again**, which has a

**1/6 chance**of hitting another

**Special**, which is

**2 + the average value of the die**

**again**, and

**again**, and

**again**; you see the problem here. The

**Special**is more tricky to evaluate since it includes itself in its own definition, and we know that this can be

**up to infinite in value**.

With that in mind, how can we get an

**average**

**value**when

**one of the possibilities is infinite**? Well, the chance you roll

**multiple Specials**in a row is highly unlikely. At some point the probabilities are so insignificant as to be inconsequential, they won’t affect the final result in any meaningful way. To solve this we can use a

**geometric**

**series**, but that involves maths that I once understood but now can no longer remember. Never fear though, the internet is here! Apparently, the

__average value of an exploding die simplifies to:__**To put that into perspective, it then averages**

*more than a die with*

**three 2 sides**, such as

**AtG**

**Jyn Erso**, who has an

**average value**of

**8/6:**

**1.3333**, but less than a character with a

**3 side**and

**two 2s**, like

**SoH Maul**,

**9/6:**

**1.5**.

Evidently,

**Exploding dice**really adds a good amount of

**average value to a die**, in this case adding effectively

**1.4 to a single side of the die**which… hey hang on, that

**added value**is on the

**Special**! The logic checks out, the

**Special**has a value of

**2 + 1.4, 3.4**. We can also understand it in this was:

**Every time you resolve the Special**you are functionally getting another

__and__it explodes**die**

**roll**, which has an average of

**1.4**.

**3. ANYDICE:**

Of course, you can’t trust everything you read on the internet. Especially when it involves maths that gets complicated enough that someone, like me, can easily make an error. The thing about

**dice rolls**and probability though, is that it’s very easy to simulate. I used a handy program on the internet called

**ANYDICE.com**that allows you to

**simulate dice rolls**,

*including*

**exploding dice**. You can check the code

**here**, make sure to select the

**Summary tab in data**when you output, as those are the values we care about. Yes, I know I could have written my die a lot simpler, but this is to illustrate the point!

**The result is that on average, the die has a value of 1.4**. Perfect! Our

**formula**and our

**simulation**line-up!

**4. THE CASE OF BLACK ONE:**

Now, you might be worried that this is already getting a little too divorced from reality, because after all:

**These dice explode only if specific conditions are met**. Alright, I’ll bring it down a little and talk a little bit about a card I really liked and used back before it got rotated out, a card I swore was effective, and yet no one else seemed to play. The original,

**Awakenings**

**BLACK ONE**.

**Black One**is a vehicle that

**costs 4 resources**and ONLY has a

**2 Ranged**and a

**3 Ranged damage**

**side**. Naturally, you want your

**dice cards**to be cost effective, and once

**Legacies**dropped

**Black**

**One**was competing in a world with

**T-47 Air Speeders**,

**Fang**

**Fighters**and

**Y-wings**! The

**Fang**

**Fighters**in particular was a stern competition having

**identical**

**damage**

**sides**,

**2**and

**3 Ranged**, but costing just 3 resources. All the competing vehicles each have

**maximum**

**potential**

**damage**

**ratios**

*equal to or higher than the resources spent playing them*; in an ideal world the

**FANG FIGHTER**is

**3 damage**for

**3 resources**,

**T-47**

**AIRSPEEDER**is

**2 damage**for

**2 resources**and so on. Also, the more your

**resources**are spent on a

**single die**, the weaker you are to removal; that card that removed your

**Black One**stopped you using

**4 resources worth of die**for this round! There was however one thing

**Black**

**One**did that all other vehicles didn’t: It exploded!

**The die that is**, not the vehicle.

**Black One**has a

**Special**that does

**1 damage**

*and*

**explodes**

**the**

**die**.

*Without*the

**exploding**

**die**the average value of

**Black**

**One**would be:

(3+2+2+1+1+0)/6 =

**1.5**

Meanwhile the

**Fang Fighter**is:

(3+2+1+1+1+0)/6 =

**1.333**

The difference in value between the 2 dice is 1/6, for a full resource. Just not worth it! Counting in the

**exploding**

**die**however,

**Black**

**one**becomes:

1.5 / (1 - 1/6) =

**1.8**

That’s almost

**a difference of 3/6!**The

**Black**

**One**

**Special**has a value of

**2.8**, meaning it has a

**3 damage side**, a

**2 damage side**and a

**2.8 side**. Does that make it worth the

**resource**? Well I don’t think we quite have enough information to decide that.

The thing is, this game also includes

**discard-to-rerolls**and

**Focus**

**sides**. The

**Fang**

**Fighter**is actually

**a pretty weak die value-wise**, but it’s ability to have a

**1 for 1 damage ratio**(1 damage for each resource spent) can really help

**maximise**

**the value**of

**Focus**

**sides**. The

**best**

**support**

**decks**typically have a lot of

**Focus**

**sides**, and aim to ignore this whole

**average**

**value**nonsense, instead forcing the

**maximum**

**value**

**side**, and there

**Fang**

**Fighter**has

**Black**

**One**beat, as they both have a maximum side of 3. Or does it?

**5. MR. POSTER BOY:**

Hah, and you thought my earlier rambling had no point to it!

**YODA**is perhaps heroes' biggest poster-boy for

**focus**

**shenanigans**, with his

**Special**

**sides**allowing him to

**focus**

**dice**and so

**Special**

**chain**. The deck I have been alluding to many of you would know as

**Drive-by-Shooting**, taken to last year's

**Worlds 2018**by

**Joe Colon**

*aka*

**HonestlySarcastc**from

**TheHyperloops**. This was my version of that deck:

This deck, similar to

**Joe Colon's original deck**, was designed to have a

**good amount of focus**and

**rerolling**in order to

**maximise the value**of our large number of dice. However, I would often have situations where I had a lot of

**excess**

**focus**, and

**Black**

**One**helped resolve that issue. Since

**Black**

**One**has a

**non-conditional exploding die**(you always get to resolve it and reroll it), you can think of

**resolving any**

**Focus side**to turn its

**die**to a

**Special**as if it is in fact

**. It’s not great, especially in a situation where you’re turning**

__converting focus into damage at a 1:1 ratio__**Black**

**One**from a

**2**or a

**3 Ranged damage side**to its

**Special**just to "benefit" from the

**Focus**

**side**. However, if we assume there will be

**no rerolls**or

**focusing**afterwards, then as the

**Black One Special**has an average value of

**2.8**, that isn’t particularly good seeing that the alternative would be a

**3 Ranged damage side**!

**6. THE MISSING LINK: REROLLS**

There’s been

**a missing link to the equation**I have set out earlier however, NAMELY,

**the more you decide to reroll**,

*the higher your*. This is obvious, when you look at it this way: If we reroll every time we hit a

**average****value**becomes**Blank**then obviously

**we are increasing a die's expected average value**. This is where the maths gets really complicated, so instead I’m going to go the simulation route for this.

Let’s assume that on a

**die**there are

**only 2 sides**we are willing to immediately resolve. For the

**Fang**

**Fighter**that would be its

**2 Ranged**and

**3 Ranged damage sides**, for

**Black One**it's the

**Special**and

**3 Ranged damage side**. Any other result we will

**reroll once**, and then we take what we can get.

Do you see that? Calculating with

**one**

**reroll**the

**Fang**

**Fighters**

**expected**

**value**increased by

**0.39**(from

**to**

**1.33****), whereas the**

**1.72****Black**

**One's**increased by

**0.44**(from

**1.8**to

**2.24**). The more rerolls, the greater the average becomes with

**exploding**

**dice**

*over*

**non-exploding dice**.

**If we decide to take the 2 Ranged**instead of rerolling, we actually go up to

**2.28**. Fancy that, I should not have been rerolling those

**2 sides**when I only had a

**single**

**reroll**and

**no**

**focus**.

**7. ADDING FOCUS SIDES:**

Speaking of

**Focus**

**sides**, let’s add them into our model. This program assumes we have

**1 Focus**

*and*

**1 reroll**available to us: If we don’t hit anything with our

**reroll**we instead

**Focus**into the

**maximum**

**side**. If we do hit our maximum side though, then we’ll have a

**Focus**

**left**

**over**that can be used to gain a value of ... let’s say 1.

**For a Fang Fighter the average value becomes: 3.28**. A consistent value of 3, but sometimes we get to keep our

**Focus**.

**Without rerolls, our average value becomes 3.17**, the reroll really doesn’t do much, as we are turning to the

**3 Ranged damage**anyway, there’s just

**an extra 1 in 6 chance**we get higher than a

**3 Ranged damage**because we get to keep our

**Focus**.

**For Black One the average value becomes: 3.72**. Why is this increase so significant? Well,

**the more rerolls the greater the chance of rerolling into a Special**,

*meaning*

**more**

**explosions**, and the

**Focus**inherently rerolls.

**At this point we actually go down to 3.64**if we take the

**2 Ranged damage**when it’s immediately offered, and with

**no**

**rerolls**at all we

**go down to 3.40**. A stark contrast to

**Fang**

**Fighter**, which benefits with a reroll by only

**0.11**more compared to

**0.24**.

With a

**second**

**Focus**and a

**Reroll**?

**4.28**vs

**4.98**. For the

**Fang Fighter**this makes logical sense, we cannot do anything more with

**2 Focuses**than we could with

**1**, so the total value only increases by

**1**, the value of the

**Fcous**. On

**Black**

**One**however

**every**

**extra**

**Focus**has a

**total value of 1 + (1/6) chance**of

**another**

**explode**by way of a

**Special**. At this point it is correct to only ever take

**3s**, otherwise

**Focus**to the

**Special**and resolve. I had a play around and

**4.98**was the best I could get, which means an extremely likely chance to be

**able to convert 2 Focuses**and a

**Reroll**into a

**5 off of the back**of a

**Black**

**One**. It bears repeating however the assumption that

**Focuses**can always go somewhere else for a value of 1, and only 1, not focusing a

**Blank**into a

**2**or anything else.

**8. ONLY SLIGHTLY WRONG:**

**, is a**

Now the real question

Now the real question

**0.7 value**

**difference**

*worth*

**a**

**full**

**resource**to play the

**vehicle**? Well remember how earlier I mentioned the

**1:1 ratio**? Well now, we can see that in our scenario here of

**1 Reroll and 2 Focuses**, the

**Fang**

**Fighter**has an average value of

**4.28**, costing

**3 resources**, as opposed to

**Black One's 4.98**, costing

**4 resources**. By this measure,

**Fang**

**Fighter**is still

**0.3 resources**

**more**

**efficient**. I guess I might have been wrong after all. Still though, it’s better than people give it credit for! Under the right circumstances it is only slightly inefficient compared to the

**Fang**

**Fighter**!

**9. REY?**

So, back to

**Rey**. I mentioned

**Yoda**for a reason dammit, and that’s because his

**Special**is a perfect fit to provide Rey with the shield necessary to make her Special explode! When you turn one of

**Rey's dice**to a

**Special**after grabbing a

**Shield**with a

**Yoda**

**Special**, you are getting an

**average value of 3.4**from

**Rey's die exploding**. This is because the

**explodey**

**bit**is worth

**2**instead of

**1**, massively increasing

**the**

**value of focuses**(in this instance

**Yoda's Special**) used on it.

What about with the

**rerolls**and a

**second**

**Focus**?

Well, from

**Rey's die**that gets

**aggressively**

**rerolled**and

**focused**,

**6.60**! Of course, we are having to spend part of a

**Yoda Special**that could have been something else, like a

**Resource**; if we include that in the calculation then it drops to a ‘humble’

**5.59**. It’s unlikely we get

**2 Yoda Specials**though, more likely we only get 1 and instead have an

**extra**

**reroll**because of

**Rey's Power Action**, bringing us to

**4.49**or

**3.59**if we include the

**opportunity**

**cost**(the fact that it could have been something else instead of a

**Shield**) of

**Yoda**

**focus**. Still though, when

**most character dice**have a maximum of

**2**or

**3**, then

**Rey**is quite a cut above most of the other characters in the set, provided she can always meet her

**explosion**

**condition**, such as by having

**Yoda**as her partner. Pity it’s all in

**Shields**...

Since you got this far, it would be remiss to give you all this theoretical knowledge without a deck to apply it to, so here is a

**Reyda**

**deck**that I have been tinkering with lately, attempting to abuse those

**exploding**

**dice**as much as possible. Remember, be greedy, and

**HAVE**

**FUN**

**EXPLODING**.

**10. DECK LIST:**

**8. FINAL THOUGHTS:**

Thishasbeenme,

Thishasbeenme

**Gameslayer989akaNooBrainerthankyouforreading**, I hope you all learnt something from this, because I certainly did.

**AnyDice**is an awesome program and I highly recommend you try it out for yourself. And thank you to

**YOUR Destiny**for hosting and editing this article, because oh boy you guys did not want to be stuck reading the first draft :P