Monday, March 24, 2014

Further Examination of Karsten Frank's Analysis

In the previous post, as well as in Karsten Frank's analysis, there is a subtle nuance that we have to touch upon. In both his article and in my initial post we assumed that we can read off the table the number of lands that is required for a spell like Anger of the Gods by looking at double-casting spells on turn 3. But, the table doesn't show that -- it shows the number of colored lands required for casting a CC spell on turn 3, assuming we have enough lands (i.e. 2 lands on turn 3). This is not the same!

It turns out that casting a 1CC spell on turn 3 requires less colored sources than casting a CC spell on turn 3. This sounds absolutely crazy at first, but remember the first assumption, that we have enough lands to cast them in the first place. Consider for a second, that we want 0% color-failure in a 24-land 60-card deck. To have no color failures on turn 2 means that we need to have 24 colored mana. On turn 3, you might think that we can allow non-colored mana, but that is wrong since we are comparing against the chance of having enough lands and that includes the chance of having only 2 lands on turn 3 - in which case there is a non-zero chance that one of them is the off-colored mana source. 

In other words, the pre-condition "have enough lands to be able to cast the spell" over-inflates the colored-land tables that we have presented, and it may be more useful to make a table consisting of the required number of lands in order to avoid color-failures for each of the consecutive spells. So, for instance, a table comprising C, 1C, 2C, 3C, 4C, 5C and 6C for single-colored spells and a table CC, 1CC, 2CC, 3CC, 4CC, 5CC for double-colored spells and so on, given at the first turn you can cast them:

Spell TypeC1C2C3C4C5C6C
90.00%1413119876
As you can see, this is quite different from the previous table. For double-colored spells:

Spell Type-CC1CC2CC3CC4CC5CC
90.00%-201715131110

And finally for triple-colored spells

Spell Type--CCC1CCC2CCC3CCC4CCC
90.00%--2219171513

From this, we can revise our numbers for Anger of the Gods, which requires 17 out of 24 red sources, and Cryptic Command that requires 19 blue sources out of 24.

For completeness, here's a few variations:

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16/40    --    1    2    3    4    5    6    7
     C    9    8    7    6    5    4    4    3
    CC   13   12   10    9    7    7    6    5
   CCC   15   13   11   10    9    8    7    6
  CCCC   16   14   12   11   10    9    8    7


17/40    --    1    2    3    4    5    6    7
     C   10    9    7    6    5    5    4    4
    CC   14   12   10    9    8    7    6    5
   CCC   16   14   12   11    9    8    8    7
  CCCC   16   15   13   12   10    9    9    8


18/40    --    1    2    3    4    5    6    7
     C   10    9    8    7    6    5    4    4
    CC   14   12   11    9    8    7    7    6
   CCC   16   14   13   11   10    9    8    7
  CCCC   17   15   14   12   11   10    9    8

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22/60    --    1    2    3    4    5    6    7
     C   13   12   10    9    7    6    6    5
    CC   19   16   14   12   10    9    8    7
   CCC   21   18   16   14   12   11   10    9
  CCCC   21   19   17   15   14   12   11   10


23/60    --    1    2    3    4    5    6    7
     C   14   12   10    9    8    7    6    5
    CC   20   17   14   12   11   10    9    8
   CCC   22   19   16   15   13   12   10   10
  CCCC   22   20   18   16   14   13   12   11


24/60    --    1    2    3    4    5    6    7
     C   14   13   11    9    8    7    6    5
    CC   20   17   15   13   11   10    9    8
   CCC   22   19   17   15   13   12   11   10
  CCCC   23   21   18   17   15   14   12   11


25/60    --    1    2    3    4    5    6    7
     C   14   13   11    9    8    7    6    6
    CC   21   18   15   13   12   10    9    8
   CCC   23   20   18   16   14   13   11   10
  CCCC   24   21   19   17   16   14   13   12


26/60    --    1    2    3    4    5    6    7
     C   15   13   11   10    8    7    7    6
    CC   21   18   16   14   12   11   10    9
   CCC   24   21   18   16   15   13   12   11
  CCCC   25   22   20   18   16   15   13   12

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