User:MagmaHound/Testing

nerdy stuff alert

There are over 1.9 billion possible decks in Clash Royale. But if you picked a random deck amongst these, what would be the expected average Elixir cost? As in, what is the average of the Average Elixir Costs of all possible decks?

There are 58 cards in Clash Royale, each with their own constant Elixir cost between 1 and 9 inclusive when it comes to calculating AEC. So, to start out, we list down the number of cards with each particular Elixir cost.

* Remember that the Mirror is treated as a 2 Elixir card in AEC calculation.

Instead of using brute force, we'll use logical reasoning to figure out the average of all Average Elixir Costs of all decks.

So, we'll start out with a simple example, using only 6 cards and a 3-card deck, with the cards having a cost of 1, 2, 3, 4, 5 and 6 respectively. That setup only has 20 unique decks, and each of the decks is unique in the sense that the Elixir costs of the three cards are unique. You could list them all down if you want to, but in case you don't have paper or don't want to do so, here they are:

Notice that, in the list of 20 possible combinations, each of the six numbers appears an equal number of times i.e. 10 times.

Now, for this next part, I'll introduce you to the permutation function. If you haven't learned combinatorics, the permutation function P shows you how many ways are there to pick a certain number of objects with regards to order, and it can be written as nPr.

For inputs n and r, nPr is equal to $$n! \over (n-r)!$$. What we want in this small example is 6P3, which equates to 120. Out of these 120, any particular number appears 20 times in the first position, 20 times in the second position, and 20 times in the third position, for a total of 60 appearances in the list of 120 overall. Eliminate repetition which are permutations of the 20 combinations above (for example 1,3,2 being a repetition of 1,2,3), and you are left with the 20 combinations you see in the list above.

Since we have seen that each number appeared 10 times, the total Elixir cost of the 60 cards within these 20 decks is equal to 210, for an average total cost of 10.5 Elixir for each 3-card deck. Divide this figure by the number of cards in each deck (3) and you will get an average of all Average Elixir Costs which is equal to 3.5 Elixir.

Now let's expand this to full scale, our 58 cards and (mostly) 8-card decks. Again, let's number these cards 1 to 58 for ease of identification.

You could go on listing all possible combinations of 8-card decks with the numbers 1 to 58, but since there's 1,916,797,311 of them, you would probably succumb to old age because this would take at least 60 years if you managed to enter one combination per second. Not to mention the extra labor you'd put in trying to sum them all up.

So, we'll use permutations and logic (again). Taking into account the order of the cards, there will be 58P8 = 77,285,267,579,520 possible ways to do this. In 1/58 of these ways, card 1 will occupy the first slot. Similarly, 1/58 of these ways will have card 2 in the first slot, and so on. Each card will make 1,332,504,613,440 appearances in each slot in the 77 trillion ways while regarding order, and 264,385,836 appearances in the 1.9 billion unique decks.

Thus, we can calculate the total Elixir cost of all 15-ish billion cards in the 1.9 billion decks. It turns out to be the sum of the costs of all 58 cards (243) multiplied by the number of appearances made by each card (264,385,836) which becomes 64,245,758,148.

But we are not done! Don't forget the seven "special decks" that have less than 8 cards, which you have surely used at least two of during the tutorial! They are:

These seven decks have a total Elixir cost of 173 across 48 cards. It's quite negligible actually :P

So the total cost of all 1,916,797,318 decks is 64,245,758,321. The average of that is about 33.51724, and averaging that over 8 cards would give about 4.1896551684 Elixir. You could ignore the seven decks I mentioned earlier and still be correct to 7 decimal places :P as the average of Average Elixir Costs of the 1,916,797,311 decks is 4.1896551724 :D

Credits to SuperRobot9338 for helping me with this :P

This is a list of most cards comparing their statistics. These statistics are correct at Tournament level.

Some cards, such as spawners are not included here due to difficulty in comparison. All troops here include swarm cards, which may have misleading data as the statistics are for a single unit. Multiply the individual stats by the appropriate troop count for a more reliable figure. Only the damage per second column is multiplied here.

Cards marked with * do not attack troops. Sub-troops and death damage are included here, as separate rows.

Cards highlighted in green background are incapable of attacking air units (which are highlighted in blue).

The ultimate purpose of this is simply reference. It could make a good starting point for a HTK calculator.

NOTE: Some cards' stats will have changed in the September 19 balance and thus the stats shown here are approximate. Once confirmed stats for the tournament levels have been found, this table will be updated as necessary.

at level 9, at level 7,  at level 4,  at level 1.

** The Poison does 550 damage on troops every time it is cast. Unlike other spells, it doesn't happen all at once.

Card ratings. Generic enough aren't they? There are quite many out there, but I'm trying a completely different approach. And while I may not have played the game in a long while, I would still know quite a bit about these cards.

A note about Traits
Certain troop cards will have traits.
 * Demoman - Prioritizes buildings over anything else.
 * Swarm - Attempts to overrun the enemy.
 * Splash - Deals splash damage.
 * Last Hurrah - Does death damage.
 * Flying - Flies over the Arena and cannot be hit by some troops.

Health, Damage and Range
The health, damage and range of troops are rated on a five star scale. The stats of troops are taken from Tournament levels i.e. level 9 Commons, level 7 Rares, level 4 Epics and level 1 Legendaries. Note that these stats may be a bit misleading since the scales are hard limits, and health and damage represent the potential health and damage of each card.

Health: (individual troop health multiplied by troop count)
 * One star - less than 400 HP
 * Two star - 400-800 HP
 * Three star - 800-1500 HP
 * Four star - 1500-3000 HP
 * Five star - over 3000 HP

Damage: (individual troop DPS multiplied by troop count)
 * One star - less than 100 DPS
 * Two star - 100-200 DPS
 * Three star - 200-300 DPS
 * Four star - 300-400 DPS
 * Five star - over 400 DPS

Range:
 * One star - melee range
 * Two star - 0-2 tiles
 * Three star - 2-4 tiles
 * Four star - 4-7 tiles
 * Five star - over 7 tiles

=My Speculation on Max+1 Stats= As of September 19 Balance Update, a max Mirror will be able to mirror cards 1 level higher than max. So what does this entail? Statistics of these new level 14 Commons, level 12 Rares , level 9 Epics and the returning level 6 Legendaries will have to be there somewhere, so this is a list of estimates.

Assumptions

 * Health and damage are increased by 10% rounded down if necessary, or to the nearest 10 if applicable, as this appears to be the protocol for leveling cards. DPS as a derived quantity will also be rounded down to the nearest integer.
 * Duration of non-damaging spell effects are increased normally (i.e. linearly).
 * This also provides for balance changes coming in the same update.

Troops and Defenses
Notes:
 * 1) The Skeleton Army's stats are very likely to be confirmed as a Skeleton Army of level X spawns in level X Skeletons. Level 9 Skeleton Army is very likely to spawn level 9 Skeletons, which coincidentally is tournament standard, but we're talking about maxed out stuff here.
 * 2) Witch was added in the (upcoming) "Spawners" section.

