The scoring system for the 3rd Mix is very different. In this system, high scores are not determined by large combos, and the maximum score for any one song is 10,000,000 This is generally believed to be the most *fair* system of scoring, but many would rather totally annihilate their opponent with an exponentially higher score than for it to come down to a few points.

First, we need to calculate the base step score P:

Let N = Total number of steps in song
P = int(1,000,000 / (N * (N + 1) / 2))

Also, there is a "bonus" for the final step of a song, that evens out the score to 10,000,000 if you get all Perfects. This bonus is calculated as follows:

Q = 10 * [1,000,000 - int[1,000,000 / (N * (N+1) / 2)] * (N * (N+1) / 2)]
*Note: Only awarded on the final step. If you get a Perfect on the last step, you get all of Q. If you get a great, you only get half of this. If you get a good or worse, you get nothing.

Let us call this new value S

If this is the 2nd step, multiply S by 2,
If this is the 3rd step, multiply S by 3,
and so on...

Here is an illustration with a 4 step song (not that any actually exist):

P = 1,000,000 / (4 * (4 + 1) / 2)
= 1,000,000 / (20 / 2)
= 1,000,000 / 10
= 100,000

If the step is "Perfect", S = 100,000 * 10 = 1,000,000
If the step is "Great", S = 100,000 * 5 = 500,000
If the step is "Good", S = 100,000 * 0 = 0

The bonus, Q, is 10 * (1,000,000 - int[1,000,000 / 10] * 10) = 0 points for this song. Bummer.

So, if we had all "Perfect":

Step 1: Score = (1,000,000 * 1) = 1,000,000
Step 2: Score = Previous Score + (1,000,000 * 2) = 3,000,000
Step 3: Score = Previous Score + (1,000,000 * 3) = 6,000,000
Step 4: Score = Previous Score + (1,000,000 * 4) + Bonus = 10,000,000

Let's Do it again with a few "Greats" thrown in:

Step 1 (Perfect): Score = (1,000,000 * 1) = 1,000,000

Step 2 (Great): Score = Previous Score + (500,000 * 2)

= 1,000,000 + 1,000,000
= 2,000,000
Step 3 (Great): Score = Previous Score + (500,000 * 3)
= 2,000,000 + 1,500,000
= 3,500,000
Step 4 (Perfect): Score = Previous Score + (1,000,000 * 4) + Bonus
= 3,500,000 + 7,000,000 + 0
= 7,500,000

However, let me illustrate another situation:

Step 1 (Great): Score = (500,000 * 1) = 500,000

Step 2 (Great): Score = Previous Score + (500,000 * 2)

= 500,000 + 1,000,000
= 1,500,000
Step 3 (Perfect): Score = Previous Score + (1,000,000 * 3)
= 1,500,000 + 3,000,000
= 4,500,000
Step 4 (Perfect): Score = Previous Score + (1,000,000 * 4) + Bonus
= 4,500,000 + 4,000,000 + 0
= 8,500,000

See? You will get a higher score if you get your "Perfect"s at the end as opposed to the beginning, but if you have been studying the above equation, and are proficient in basic algebra, you already knew this (^_^)

Also, "S" is incremented only as step judges are determined. If you miss one step, and get a Perfect on the next BEFORE the first one is judged as Missed, you receive points (and a raise in combo) according to what S would have been had that second step come first.

Your stage Grade is calculated as follows:

Each step has a value of two "Dance Points" assigned to it. When you add them all up, you get the maximum number of possible "Dance Points". (Equal to 2 * N)

Your earned "Dance Points" are calculated as follows:

A "Perfect" is worth 2 points
A "Great" is worth 1 points
A "Good" is worth 0 points
A "Boo" will subtract 4 points
A "Miss" will subtract 5 points


SSS = All Perfect
SS = At least 80% Perfect and the rest Great
S = Less than 80% Perfect and the rest Great
A = At least 80% of maximum Dance Points
B = At least 64% of maximum Dance Points
C = At least 50% of maximum Dance Points
D = Less than 50% of maximum Dance Points
E = Failed
*Note: A grade of SSS through S override any other grade. If you have less than 80% of maximum Dance Points, but a Full Combo, the S grade overrides the B. Likewise, if you do not have a Full Combo, the highest grade you can get is an A.

Grading system derived by Taren N.

Just like 1st and 2nd mixes, you get a score based on combo, using the familiar int(C/4) formula. You also get a bonus for your stage number. By this formula, you can tell there is more emphasis on stage count rather than combo length.

Let C = current combo, including step.
Let S = current stage number
Let M = int(C/4) = temporary combo factor
Let E = int((S-1)/8) + 1 = temporary stage exponent
Let L = (S^3 - S^2) = temporary stage multiplier. (If stage is 1, make L '1' instead of '0'.)

Using these temporary variables, the formula determining the value of each step is as follows:
Perfect = (M^E + 1) x 300 x L
Great = Perfect / 3.

Substituting the equations back in, it is:
Perfect = [int(C/4)^(int((S-1)/8)+1)+1]x300x(S^3-S^2)
Great = Perfect / 3.

If you break your combo, rather than starting from scratch, a provisional combo multipler is used. This is very useful, especially if you have a big combo and a bad break. Let's say (M^E + 1) is your combo multiplier, called M' (M-prime). This steps up by one every four combo, and even faster in later stages. If you break your combo, a provisional combo multiplier is used, set at int[(M' + 1) / 2]. This provisional multiplier is used until M' exceeds P. That way you don't start from scratch. If you break your combo while M is less than P, P is reassigned to int[(P+1)/2] until M' exceeds that.

Examples:
I started stage 2 with a 69 combo. I got a perfect on the next step, so I get 21600 points.

C = 70
int(C/4) = 17
... ^ (int(S/8)+1) = 17
... + 1 = 18
... * 300 = 5400
... * 2^2 = 21600 points.

M' was int(81/4)^1 + 1 = 20 + 1 = 21.
P will be int(21 + 1) / 2 = 22 / 2 = 11.
11 * 300 = 3300 * 2^2 = 13200 points.

M' was 1, so I'll use P since it's higher.
P was 11.
P will be 11 + 1 = 12 / 2 = 6.
6 * 300 = 1800 * 2^2 = 7200 points.

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