This page contains the first part of the reasoning behind the KirriCorp League's restriction of all decks to 40 cards exactly. Specifically, this page contains the mathematical explanations of the effect that allowing more than 40 cards has on the Duel Masters™ game with regards to the easily available deck-search methods inherent in the game. The explanations have been slightly simplified for ease of understanding. Feel free to test anything not directly mentioned, but note that if it was rather basic probability mechanics, it was probably only omitted for the sake of brevity.

The first thing that should be noted is that this deck is less than functional, but can approximate the level of an average deck with a little luck on the draw. The deck contains no recursion mechanic, no spell based draw-engine, no way of returning cards from the mana zone, and no way of sending a card from the top of its deck or from grave to mana. It can play only what it draws or searches out. This is very important.

The deck contains fifteen 'singles', each chosen for its ability to deal with a different situation that the deck might find itself in. It is not adapted for attacking swiftly, but generally can begin an assault on the sixth turn. The primary capability of the deck is its large number of search cards. The purpose of this dissertation is to prove that the increase of effectiveness of the search mechanic is large enough to be considered aberrant to the game's original design.

The KC league, like the Japanese version of the game, restricts decks to a maximum of forty cards because the Duel Masters™ game, unlike many other TCGs, was designed with this in mind. This is demonstrated in the larger amounts of cheap draw and easy search cards that are available in the Water and Nature Civilizations especially. Today, we tackle the fundamental mathematics behind why a game that allows more than 40 cards would not have such search and draw easily available, using the above deck.

The first opposing deck will be assumed to be a Light deck depending on powerful Blockers, Petrova, and Diamond Cutter. We will therefore define 'effectiveness' of Calculation deck against this opponent to be 'summoning Scarlet Skyterror by turn 9 and 'ineffective' as all other scenarios. We will also assume that the Diamond Cutter Terminus does not attack at all until it is ready to strike for the win.

In the deck shown, the chance of the Scarlet Skyterror being shieldlocked is 12.5%, or 1 in 8. By the given scenario, this means that the deck absolutely will be ineffective in 12.5% of all its duels against Diamond Cutter Terminus, since it becomes impossible to draw the Skyterror.

On the 9th mana, it is assured that at least 18 of the deck's 40 cards are no longer in the deck, leaving 22 or less untouched. If no search was available, this would mean approximately another 20/40, or 50% chance, that the deck would be ineffective, bringing the total ineffectiveness to a full 62.5%.

The deck does, however, contain 8/40 cards capable of searching out a Scarlet Skyterror. The chance of none of these cards, including the Skyterror itself being available by turn 8, is given approximately by the following equation:

(31!/18!) / (40!/27!) = 0.01714

This changes the effectiveness of our deck from 37% effective to between 88% and 98.3% effective, depending on shieldlocking of varying cards, including the Skyterror itself.

Now, let us compare the effectiveness of a 41 card version of Calculation, containing 2 Scarlet Skyterror, to a 40 card version containing 2 Scarlet Skyterror and 1 less Rumbling Terahorn.

The chance of both Skyterrors being shieldlocked in the 40 card and 41 card versions are minimally different, but it should be noted that adding the second Skyterror does drop this chance to less than 1% in both decks. By adding one card. The ratio change achieved through this does not change the equation for the other ability very much in the 40 card deck, it remains approximately 2% chance of not drawing any search or Skyterrors, but now the variance in effectiveness due to shieldlock is greatly reduced, putting effectiveness at between 98% and 99%.

The equation given for the 41 card deck is given approximately by the following:

(31!/18!) / (41!/28!) = 0.01171

This change is also very minor, lowering the chance of drawing no search or Skyterrors closer to 1%, but still not making any large difference directly.

For a 42 card deck, the equation would be close to the same yet again:

(31!/18!) / (42!/29!) = 0.00808

With another minor reduction. This demonstrates nothing more than a higher chance of obtaining the Skyterrors based on the obvious solution of, clearly, adding more Skyterrors. Now, let us contrast the probability of not getting the single available Bazagazeal Dragon out of the same decks, on the same turn, rather than the Scarlet Skyterror. (We are now assuming that a different opponent is being faced, and Bazagazeal is somehow now the definition of effective).

In the original 40 card version, the chances of not getting Bazagazeal are the same as the chances of not getting the single Scarlet. In the 40 card version where we have replaced one Terahorn with a Scarlet Skyterror, the chances of being unable to get Bazagazeal are now:

(32!/19!) / (40!/27!) = 0.02887

This gives the effectiveness of our deck in getting the Bazagazeal Dragon to between 88% and 97.1% effective, depending on shieldlocking of varying cards, including the Skyterror itself. However! In the 41 card version, the chances of being unable to get Bazagazeal are:

(31!/18!) / (41!/28!) = 0.01972

This seems like a very small change. Less than 1% difference, and it is. Let us continue to the situation of a 40 card deck that contains 3 Skyterrors, 1 less Crystal Memory, and one less Rumbling Terahorn, attempting to get the same Bazagazeal Dragon:

(33!/20!) / (40!/27!) = 0.04763

This is almost a 3% decrease in the effectiveness of the deck in getting Bazagazeal 'in time', from the original, and this does not even factor in the idea that the card may be shieldlocked. Let's look at the 42 card version of the deck that still has all its search cards, but has the two extra Skyterrors:

(31!/18!) / (42!/29!) = 0.02246

This deck is 2% less likely to be unable to get its Bazagazeal Dragon than its 40 card counterpart with an equal number of Skyterrors. Now, to avoid getting too bogged down in this, let's just jump straight to the final comparison. First, the chances of not getting either Bazagazeal or Scarlet out of a 40 card deck that has 3 of them each, but only 2 Crystal Memory and 2 Rumbling Terahorns:

(33!/20!) / (40!/27!) = 0.04763

We have seen this number before. This has been previously calculated, and it is oddly(it isn't really odd), the same as the chances of not managing to get the single Bazagazeal Dragon, with more search cards. What about the chances of not getting either one out of a 44 card deck with all the search cards available?

(33!/20!) / (44!/31!) = 0.01104

The more search cards available, the higher the chance of getting exactly what you want. A simple conclusion. But now let's look at something else altogether. The chance of being able to get the Bolmeteus Steel Dragon out of these same two decks. Firstly, 40 cards, 3 Skyterrors, 3 Bazagazeal, 4 search:

(35!/22!) / (40!/27!) = 0.12269

Now hold on. This implies that the deck will fail to get its Bolmeteus Steel Dragon almost a quarter of the time, either because it is shieldlocked, or because it draws neither Bolmeteus nor any search cards. One was better off with a lot of search and only one of everything. Considerably so. What about the 44 card deck?

(35!/22!) / (44!/31!) = 0.02844

This deck still has to worry a bit about the card being shieldlocked, since the chances of that do not change by too much, but look at that wonderful difference between the chances of it not being obtainable at all! If we add a second Bolmeteus, it brings the shieldlock likelihood down to under 1% as well! In fact, let's just max out the search and put in two of everything we only had singles of before! That way, we practically eliminate shieldlock as an obstacle to effectiveness. But we also raise our deck to 55 cards. Is this going to be a serious problem? Let's see?

(31!/18!) / (55!/42!) = 0.05031

Now the deck has approximately a 93-95% chance of being able to get any creature the player desires even if the opponent never attacks. For spells, which can only be searched via Crystal Memory, the chances are:

(49!/36!) / (55!/42!) = 0.18095

Still, 82% chance of getting any spell you want is not exactly bad either. Not much different from the chances of it being shieldlocked in a smaller deck. Draw, along with searching for creatures, can only make the chances better. Adding four Energy Stream to the deck to make it 59, and perhaps one more Hulcus to raise it to 60, improves everything, functionally.

Does the deck necessarily run 'better'? That is up to the skill of the player. It is definitely going to be more consistent, and more capable of handling different opposing decks, due to the large number of different high-mana cards it can use depending on the situation. The combination of the full complement of draw and search cards, with just about every strategy that relies on them, available, does not even need to factor in the great truth that as mana approaches higher levels, draw and search begin to have less effect on the probabilities of a deck. After 8 mana threshold is reached on both sides, the player with the Aqua Hulcus in the deck is fundamentally more dangerous as long as the deck has enough high level destructive power. At worst, they will draw a new card that they cannot play until the next turn, or a draw/search card.

This is the basic mathematical reasoning for the restriction to 40 cards, but alone, it does not answer the question of exactly why draw and search power is considered so 'bad' as to require this limit. Further 'articles' in this series will touch on the imbalance of power between the game's originally balanced play mechanics, that only 40-card restriction can restore to balance. For now, it is hoped that this analysis has shown at least one thing.

Bigger decks make search (and draw) stronger.