From: exeter@lni.net
Newsgroups: rec.games.trading-cards.magic.strategy
Subject: Mana screw [Was Re: The RULE of FOUR]
Date: Sun, 07 Sep 1997 13:02:07 GMT

On 1 Sep 1997 23:02:10 GMT, edt@admin.lsa.umich.edu (Eric Taylor)
wrote:

>
>exeter@lni.net wrote:
>
>: I believe these numbers, but something is fishy.  According to the
>: Rule of Four, shouldn't 6 diamonds be ideal for 24 mana sources?  Yet
>: noone in his right mind would build a deck with 18 land and 6
>: diamonds.  I think the answer lies in mana screw. :)  I am working on
>
>The rule of four gives you 6 diamonds in a 24 mana source deck as the
>maximum number of diamonds, but the rule of four is just a rule of
>thumb.  For decks running 24 mana with a flashpoint of 4 that you want
>to use diamonds in, 4 diamonds is probably a good starting point.
>
>You'll notice from the charts in my previous post, for mana starved
>decks the rule gives too high of a number of diamonds, but for
>mana-glutted decks, the rule gives too low of a number.

I think a better idea would be to tune your deck to draw about four
land over the first four turns.  Assuming you play first, that's 10
cards of your deck.  Using C(n,r) for "n choose r", and letting L be
the number of land in the deck, we want the following to be a maximum:

C(L,4) * C(60-L-4, 6) / C(60,10) [Probability of exactly 4 lands in 10
cards].

Going through the mathematics of differentiating that monster of an
expression gets nasty, so I'll just run the numbers for L = 15-30.

Table 1:

# of Land	Probability of drawing exactly 4 land of 10 cards
---------	-------------------------------------------------
15		8.1%
16		9.2%
17		10.2%
18		11.2%
19		11.9%
20		12.5%
21		12.9%
22		13.04% <---- Point of diminishing returns.
23		13.00%
24		12.7%
25		12.3%
26		11.8%
27		11.1%
28		10.2%
29		9.3%
30		8.4%

If instead we want 3 land and a diamond in the first 10 draws, the
table gets a bit bigger. :)

For this calculation, let L be the amount of land, D be the number of
diamonds.  We want to maximise the probability of 3 land and 1
diamond.  The probability of this happening is C(L,3) * C(D,1) * C
(60-D-L-4, 6) / C (60*10).

Table 2:

# of land	d1	d2	d3	d4	d5	d6	d7	d8

15		2.3%	3.9%	4.9%	5.6%	5.8%	5.8%	5.6%	5.3%
16		2.4%	4.1%	5.1%	5.7%	6.0%	5.9%	5.7%	5.3%
17		2.4%	4.1%	5.2%	5.8%	6.0%	5.9%	5.7%	5.3%
18		2.5%	4.2%	5.2%	5.8%	5.9%	5.8%	5.5%	5.1%
19		2.5%	4.1%	5.1%	5.6%	5.8%	5.6%	5.3%	4.8%
20		2.4%	4.0%	5.0%	5.4%	5.5%	5.3%	5.0%	4.5%
21		2.3%	3.9%	4.7%	5.1%	5.2%	5.0%	4.6%	4.1%
22		2.2%	3.7%	4.5%	4.8%	4.8%	4.6%	4.2%	3.7%
23		2.1%	3.4%	4.1%	4.4%	4.4%	4.1%	3.7%	3.3%		
24		1.9%	3.1%	3.8%	4.0%	3.9%	3.7%	3.3%	2.8%
25		1.8%	2.8%	3.4%	3.6%	3.5%	3.2%	2.8%	2.4%	
26		1.6%	2.5%	3.0%	3.1%	3.0%	2.7%	2.4%	2.0%

As per earlier results in the other thread by Mr. Taylor, the point of
diminishing returns comes at 4 or 5 diamonds, in general.

Now, if we take the numbers in the first table, representing the
probability of exactly four land in the first 10 cards, and add the
largest number from table 2 (noting the number of diamonds required),
we can then obtain a working maximum value for probability of reaching
4 mana by turn 4, which I think is a better measure than 'effective
mana' for determining a deck's consistency.  Note that the speed bonus
created by the diamonds is reflected (somehwat) in table 2.

Table 3:
# of land	# of diamonds required		Table 2+Table 1
15			5			5.8% + 8.1% = 13.9%
16			5			6.0% + 9.2% = 15.2%
17			5			6.0% + 10.2% = 16.2%
18			5			5.9% + 11.2% = 17.1%
19			5			5.8% + 11.9% = 17.7%
20			5			5.5% + 12.5% = 18.0%
21			5			5.2% + 12.9% = 18.1% 
22			4 or 5			4.8% + 13.04% = 17.8%
23			4 or 5			4.4% + 13.0% = 17.4%
24			4			4.0% + 12.7% = 16.7%
25			4			3.6% + 12.3% = 15.9%
26			4			3.1% + 11.8% = 14.9%

Amazingly, the most consistent configuration appears to be 21 land, 5
diamonds, though neither 20/5 nor 22/4 are far behind.  Personally, I
like the 22/4 configuration myself.  Note that none of this considers
colour screw, Thawing Glaciers/Rowen/Land Tax/other methods of ripping
out land from the library.  I hope someone can take the time to verify
these calculations, and perhaps refine the analysis.  For example,
table 3 completely discounts the possibility of 2 land 2 diamonds in
the first 10 cards.  That kind of draw can be devastating in some
circumstances.  I'd much prefer to have either the 4 land or the 3
land 1 diamond draw.