Welcome to a special post here on Friend & Fow which focuses on an important ruling that will be applied for the Philippine AGP (August 20, 2016). I would like to note that there is no rule change that caused this, rather this post is just to clarify any confusion about Overlord of the Seven lands, Valentina.

Welcome to the blog series here on Friend & Fow which deals with the complex decisions involved in the game and explores the probabilities, and statistics behind each topic. For this week we look at the Yamata Reanimator Combo.

The plan of the deck is to play some enablers resulting in Yamata-no-Orochi finding its way into the graveyard. This allows us to then cast a reanimation spell like Book of Eibon and start beating down the opponent with at least 1600 worth of ATK damage starting on turn 3. If we are able to pump Yamata’s base attack damage to at least 500, then that would result in a turn 3 combo kill.

Yamata-no-Orochi – Necromancy of the Undead Lord – Book of Eibon

Today’s discussion is going to revolve around how to maximize the probability of casting a Book of Eibon on turn 3, with at least one Yamata-no-Orochi in the graveyard, and at least +300 of no-mana-cost attack damage pumping.

These three cards are the meat of the combo and we’d want to play as many of them as legally possible to maximize statistics. The legal restriction prevents us from reaching more than 4 of each, but it would be an interesting puzzle trying to solve how much percentage points we could gain if allowed to increase the count.

Reflect / Refrain

During the initial iteration of the combo, we had to rely on two instances of Necromancy of the Undead Lord from the graveyard. Fortunately for us the recently “fallen” Ruler serves as a much needed consistency to the combo kill. His draw-and-bottom ability allows us to look through 2 extra cards during the first two turns, helping us find the missing piece to the puzzle. And then on turn 3, he’s able to contribute one instance of the necessary no-mana-cost +200 attack damage pump. This decreases our requirement to needing at least one Yamata-no-Orochi and at least one Necromancy of the Undead Lord in the graveyard by turn 3.

Our resources and game plan are all set for Turn 3. Let’s now talk about what we can do using the 3 other mana we have available during Turns 1 and 2.

1-Cost

Prowler of Niflheim

Perhaps the most straight forward method of sending a Yamata-no-Orochi to the graveyard, this card allows us to mill (which means to put cards from the top of the deck directly to the graveyard/discard pile) the top 2 cards from our deck. Assuming that we are able to play 3 Prowler’s during the first 2 turns, we end up with 6 cards in the graveyard. If we have 6 cards in the graveyard, the probability of having at least 1 Yamata-no-Orochi and 1 Necromancy is 22.12%.

* The 22.12% value was solved based on relevant combinations from the 6 cards and the Combination formula, denoted as “nCr”. The combination formula determines the number of possible combinations of r objects from a set of n objects. To get the probability of exactly 1 Yamata-no-Orochi and exactly 1 Necromancy, we can use 4C1 * 4C1 * 32C4 / 40C6. In english that translates to, (combinations of 1 Yamata out of the 4 in the deck) multiplied by (combinations of 1 Necromancy out of 4 in the deck) multiplied by (combinations of 4 of the other 32 irrelevant cards) divided by all possible 6 card combinations from a 40 card deck. Evaluated, the formula comes to about 15%, but that just shows the probability for “exactly” one of each. Since we don’t mind extra’s, we check for “at least” by adding up for 1 Yamata and 2 Necromancy, 1 Yamata and 3 Necromancy, 1 Yamata and 4 Necromancy, 2 Yamata and 1 Necromancy, and so on.

Guinevere

While Prowler is the best one cost creature at immediately placing a high quantity of cards in our graveyard, Guinevere allows us to select one of the cards in our hand and discard it into our graveyard while drawing two additional cards to help find those combo pieces to discard. In addition, we’re able to repeat the ability across multiple turns as long as we have other resonators to sacrifice.

Assuming that we are able to activate the queen twice, this results in 4 cards drawn and 2 cards discarded. This results in just 2 cards in the graveyard, but since we’re discarding them selectively from our hand instead of relying on the probabilities of our deck the computation is actually improved. We’re not solving for the probability of having the combo in just those 2 graveyard cards but from the 13 cards we’ve had pass our hand!

In addition to the 4 cards that we draw off the Guinevere’s activation, we also consider our starting hand of 5, the 2 draws we have (assuming we’re on the play), and the 2 extra draw-bottom effects we get because we’re using Reflect as our ruler. We already know that 2 cards are the Queen and another resonator, so we look for the combo pieces in the remaining 11 unknown cards.

From the unknown 11 cards, we need to have at least 1 Yamata and at least 1 Necromancy to discard and 1 Book of Eibon stay in our hand to play. Using the same mathematical approach as above, we end up with a 37.84% chance of having the combo pieces in the graveyard and as an added bonus already have the Book in our hand. The 22.12% of the 3 prowlers does not even consider if we have the book in hand.

As we can see, discarding cards from hand is generally better than milling random cards from the library. However, the hand cards only need one discard outlet to become “live” pieces of the combo. Guinevere’s ability allows extra draws per activation, however it is limited by the number of resonators we are able to play. Only a single Guinevere would be able to activate twice (assuming one other resonator was played before turn 3), drawing 4 cards and discarding 2. Additional copies of Guinevere only provide one activation, as they would have to sacrifice themselves.

After the first Guinevere, it would be preferable to draw more Prowlers of Niflheim. Guinevere can use the prowler to activate her ability, while gaining the 2 additional cards milled into the graveyard. A good scenario is 1 Guinevere and 2 Prowler’s of Niflheim. This allows 4 cards milled directly to the graveyard, 4 extra draws and 2 cards discarded. This results in a potential of 14 unknown cards (4 + 4 + 9 from hand – 3 resonators) that can potentially fuel our combo. That’s a 58.85% chance that we assemble the full combo by playing those three cards.

Rukh Egg

The unassuming Rukh Egg doesn’t enable cards to find their way into the graveyard, but it 100% guarantees that Yamata-no-Orochi finds its way into our hand. Rukh Egg is useless if we don’t have both a way to kill it, and an outlet to discard the searched Yamata-no-Orochi. Fortunately for us the Rukh Egg synergizes very well with Queen Guinevere. Note that the chasing of the effects requires us to resolve the searching from the egg before Guinevere’s draw 2 discard 1, which is basically the rules helping us play correctly.

Imagine having 1 Guinevere, 1 Prowler and 1 Rukh Egg. Instead of having 14 unknown cards to get one of each combo piece; we can have 1 guaranteed Yamata-no-Orochi in addition to 12 unknown cards (we replace the mill 2 effect of a Prowler, with the search effect of the Egg). 12 random cards have a 59.23% chance of having at least one Necromancy and at least one Book, further increasing the success of our combo by about 1%.

2-Cost

Forty Thieves

While Forty Thieves can be considered as just a higher costed version of Guinevere, it provides much needed consistency in terms of additional effects to both draw and discard cards. The only key card in our combo that actually needs to be in our hand is the Book of Eibon. Since we’re reliant on this part of the combo, its important we dig heavily for the Book and prioritize playing effects that draw cards into our hand to find it. It’s a safe assumption to expect to play forty thieves whenever we don’t have the Book of Eibon in hand or we don’t already have a discard outlet on the table.

Card Soldier Club + Niflheim

With one Book of Eibon in hand, we can then shift our attention to the graveyard, and these two cards provide the highest mill-able card quantity at this cost. Take note that the most important number to maximize is not the amount of cards in the graveyard. Rather, we want to maximize the number of cards that have had the opportunity to enter the graveyard. We need at least one discard effect for the cards in our hand to contribute to our graveyard. After assuring that, the priority shifts to casting the card that allows us to “see” more cards.

For example, we have a turn 1 Guinevere, a book of Eibon ready in hand, and a choice between Card Soldier Club and Forty Thieves on turn 2. It would be correct to play the Card Soldier Club. This way, 4 random cards go into the graveyard, and then we have access to two Guinevere activations (sacrifice the Card Soldier and then herself) for 4 extra drawn cards and then discard 2.

This results in 8 extra cardsseen in addition to the 9 “hand” cards.

If we were to pick the Forty Thieves the forty thieves activation plus 2 Guinevere activations result in just 6 extra cards seen.

On another scenario, we have a turn 1 Prowler of Nifleheim, book of Eibon still ready in hand, and a choice between Card Soldier Club and Forty Thieves. We now actually need to look into our hand and determine if we can discard the necessary combo pieces. If we have at least one Yamata or Necromancy in our hand and its missing from the Graveyard, Forty Thieves would be the better choice.

However, if our current hand can’t contribute any missing graveyard pieces, it would be better to play the Card Soldier Club, in the hopes of hitting the 2 combo pieces within the 4 milled from his activation (10.43%).

A quick note on Niflheim, if we have a turn 1 Guinevere then it is strictly worse than a Card Soldier Club. It isn’t a resonator, so it doesn’t provide sacrifice fodder for the Queen’s activation, resulting in 2 less seen cards and 1 less discard, negating the 1 extra milled card.

Magic Stone Base

The single most important card to cast is Book of Eibon. There’s no point going through all the trouble of sending Yamata and Necromancy to the graveyard if we can’t cast Book of Eibon due to our stones. This means that we want 2 black sources by turn 3; we’ll need 9 magic stones that produce black to reach 100% certainty. We also want to maximize the amount of red sources that we have because of Queen Guinevere being the best turn 1 play for this deck. This leads us to the following mana base:

4 Magic Stone of Moon Shade
4 Magic Stone of Scorched Bales
1 Milest, the Ghostly Flame Stone
1 Little Red, The Pure Stone

Most of the time, we’ll state Fire as the attribute of choice for Little Red, but it can be called as a black source when needed. The Milest and Little Red add additional options to pump Yamata, but they can only be useful for activation on turn 4.

Cards 37-40

Unfortunately there currently aren’t any more cards in the Black and Red colors that help improve the probability of the turn 3 combo kill.

Knight of the New Moon would be a great resonator to add able to draw and discard up to 3 times, but it is a water resonator and the strain on the magic stone base actually decreases the likelihood of the combo. All other discard outlets don’t allow us to see more cards, so they don’t increase probabilities.

The remaining slots can be dedicated to shore up what happens if we don’t have the turn 3 combo kill. We can choose to increase the amount of reincarnation spells (Genesis Creation) or reincarnation targets (Susanowo, Arthur). We can also choose to add in some helpful interactive spells such as Soulhunt that benefit our plan.

I hope this was a fun and interesting look into the Yamata Reanimator combo. What topic should we analyze next week?

Welcome to Know Your FoW! It’s the blog series here on Friend & FoW that deals with the decisions involved in the game and explores the probabilities and statistics behind each topic.

For this week’s topic, we look into what can be considered as the most straightforward aspect in the game: the Magic Stone. I call it straightforward because once you have determined that you need to call a Magic Stone, there is little to no decision making involved afterwards – the stone that gets put into play is at the mercy randomization. For the former magic players, there’s no mulligan for lands or sequencing of which to play first. For the former World of Warcraft players, there’s no need to identify which card is best to row as a resource. It’s as if we have no control whatsoever.

But we do have control. Not during the actual game, nor during shuffling (hopefully you do so as truly random as possible). We have control during deck-building. For mono colored decks this exercise is simple, just make sure you build our Magic Stone Deck with 10 of your chosen color and you’re all set. The challenge comes when one needs to find the right balance for multi colored decks.

The Ratio Method

A relatively simple method in balancing stones is to sum up the total number of attribute symbols found in the casting cost of cards in your deck and proportion your magic stones according to the identified ratio. Let’s take the following deck for example:

4x Brainless Scarecrow {G} 4x Cowardly Lion {G} 4x Heartless Tin Man {G} 4x Dorothy, the Lost Girl {G} (1) 4x Glinda, the Fairy {G} {G} 4x Deadman Prince {B} (2) 4x Pumpkin Witch {B} (2) 4x Jewels on Dragon Neck {B} (2) 4x Spire Shadow Drake {B} (2) 4x Mephistopheles, the Abysal Tyrant {B}{B} (2)

In our example, one could count up 24 {G} symbols and 24 {B} symbols, resulting in a nice 50:50 ratio. That would lead us to constructing our Magic Stone deck containing equal parts Wind and Dark, 5 of each stone without any duals, or 3 of each and 4 duals. While the ratio technique is good for a quick estimate of attribute balance, it overlooks a key factor in gameplay. Indeed half of the deck needs {G} and the other half needs {B}, but all turn one plays require {G} and only needs {B} by turn three at the soonest. This leads us to my preferred method:

The Curve Analysis Method

While taking just a bit more time and effort to implement, this method paints a clearer picture of how the deck would want to operate and the Magic Stone deck would hopefully reflect that. One would need to list out his deck by “ideal turn played” and identify what color requirements are necessary at each point in the game. Again for the example deck:

Turn 1: {G}

Turn 2: {G} (1) or {G} {G}

Turn 3: {B} (2)

Turn 4: {B} {B} (2)

On turn 1, this deck really wants to play a one-cost resonator. By this point, only one Magic Stone would have been called. The likelihood of calling a stone that can play a {G} casting cost resonator can be computed by the formula:

(# of stones that can produce wind) / (# of stones in magic stone deck)

The (# of stones in a magic stone deck) would be 10 at the start of the game, so there’s nothing we can do to improve our chances by manipulating the divisor. We can improve our probabilities by modifying the (# of stones that produce wind).

In a perfect world, we’d want to reach 100% certainty, which would require 10 out of 10 magic stones to produce wind. But as we shall see, we can’t always get 100% certainty, and maybe 90% or 80% or even 70% would be enough.

In order to cast Glinda, the Fairy on turn 2, two magic-stones should be called and both need to produce {G}. Thats actually two dependent events that need to have happened: turn 1 call a stone that produces {G} and turn 2 call a stone that produces {G}. In a formula: (# of stones that can produce wind) / (# of stones in magic stone deck) * (# of stones that can produce wind – 1) / (# of stones in magic stone deck – 1)

Note the “minus ones” in the second half of the equation, caused by the calling of the first stone decreasing the magic stone deck size. Again, to reach 100% certainty one would need 10 out of 10 magic stones to produce wind. While 9 wind producing stones amounts to 80%, 8 to 62.22%, 7 to only 46.67%. If we had followed the values specified by the ratio method we’d only be able to cast the turn 2 Glinda in our hand about half of the time.

While both Glinda and Dorothy are two-cost wind resonators, the difference in (1) {G} and {G} {G} is quite significant for your Magic Stone balance. For Dorothy, the Lost Girl, at least one of the two magic stone’s called would need to produce {G}. The event consideration is slightly more complex: if turn 1 call a stone that produces {G} then turn 2 call any stone if turn 1 call a stone that doesn’t produce {G} then turn 2 call a stone that produces {G} In formula:

(# of stones that can produce wind) / (# of stones in magic stone deck) * 100% + [(# of stones that can’t produce wind) / (# of stones in magic stone deck) * (# of stones that can produce wind ) / (# of stones in magic stone deck – 1)]

Fortunately for us, another way of considering this equation would be: what is the likelihood that we get both stones that do not produce wind and subtract that value from 100%, formula:

100% – [(# of stones that can’t produce wind) / (# of stones in magic stone deck) * (# of stones that can’t produce wind – 1) / (# of stones in magic stone deck – 1)]

This time, to reach 100% certainty we only need 9 wind producing stones. Even dropping to 7 wind producing stones, still provides us with a 93.33% certainty of being able to cast the Dorothy in our hand on turn 2.

As we move up the curve, we realize that representing the probabilities in a formula and calculating them accordingly gets more and more like a homework assignment instead of preparing for a fun card game. Worry not, because we have the internet to save us. Stattrek provides an effective hypergeometric calculator that we can use for these exact purposes (Available at: http://stattrek.com/online-calculator/hypergeometric.aspx). I’ll be referring back to this calculator many times in the future. To use it for our Magic Stone calculations, make use of the following inputs:

Population size = 10 (# of stones in magic stone deck)

Number of successes in population = up to you (# of stones that produce the attribute of your choice)

Sample size = Turn # (# of stones that have been called)

Number of successes in sample (x) = (# of attribute symbols are found in the casting cost of your card)

The “>=” result is relevant for our scenario because we want to meet the casting cost requirement but don’t mind going over.

So for Pumpkin Witch one would enter something like:

Population size = 10

Number of successes in population = 5

Sample size = 3 (assuming we want to cast her on turn 3)

Number of successes in sample (x) = 1

and be 91.67% confident that we’d be able to cast her on turn 3 even with just 5 dark magic stones.

For Mephistopheles, the Abyssal Tyrant on the other hand:

Population size = 10

Number of successes in population = 6

Sample size = 4 (assuming we want to cast her on turn 4)

Number of successes in sample (x) = 2

would give us 88.1% certainty that Mephistopheles can be played with 6 dark magic stones.

Now what do we do with all these numbers?!? With the Curve Analysis Method we’ve identified that it’s almost impossible to reach 100% certainty to cast all our cards in our multi colored deck, but we can come very close to it.

Assuming all the cards in our example deck are equally as important to be played as soon as possible, I’d probably balance the magic stones as 5 Wind stones 1 Dark stones and 4 dual stones. This leaves us with an 80% likelihood to cast Glinda on turn 2 and 73.81% chance to cast Mephistopheles on turn 4, all the other cards would be at least 90% certain that we would be able to cast them as soon as possible. Without any dual stones, I’d be very hesitant to try to make this deck work because the likelihood of successfully playing either Glinda or Mephistopheles would be too low for my preference.

It’s the cards with double casting cost requirement that are the most difficult to cast consistently in multi colored decks. However, not all cards are equally important to a strategy. Nor are all cards necessary to be cast as soon as possible. Knowing the strain Glinda and Mephistopheles are putting on our consistency, we now need to ask ourselves some questions:

1. Do we really envision casting Glinda on turn 2? Maybe we only need her later and lessen the wind stones needed.

2. Is Mephistopheles relevant enough in this deck to warrant the double dark requirement? Are we willing to play him at just 73.81% certainty, or can we remove him and lessen the dark stones needed.

3. Do we really envision casting Pumpkin Witch or Deadman Prince on turn 3? Maybe we only need them later and lessen the necessary dark stone count even further.

4. Why are there Jewels on the Dragon Neck in this deck aside from making the Ratio example all nice and balanced?

By analyzing the curve and attribute requirements, we can identify cards that don’t quite fit into the plan of our deck (even though they are powerful) due to restrictive casting costs. We also get to plan out when we envision each card to be played and increase the likelihood that we can play that card when we need it.

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