Purpose:
I quite often find myself loosing to an equally skilled (or even less so) opponent at the earlier versions of the game Dance Dance Revolution. At closer analysis this is because their "combo" is usually higher than mine where as my accuracy is better than theirs (these terms are better explained in the next section) Baffled, I will attempt to dissect the old method of scoring in order to see if there lies a "glitch in the system" if you will, which would cause my scores to be significantly lower because of my inability to hold a high "combo".

How the game is played:
Dance Dance Revolution (or "the dancing video game") is a game where the player hits any of the four buttons on a metal platform in accordance with the arrows being displayed on the screen. As the player triggers the sensor, the screen will then display a judgment of their accuracy.

So as this image demonstrates, the player on the left (Player 1) got a GOOD judgment and the player on the right (Player 2) got a PERFECT judgment. The five possible judgments are:

PERFECT
GREAT
GOOD
Boo
Miss...

When a player hits two or more steps sequentially with a judgment of either PERFECT or GREAT, they have what is called a combo. However, if the player gets a judgment of GOOD, Boo or Miss..., then the combo is reset to zero.

So in this case, Player 1 has a combo of zero (not displayed) because their last step received a GOOD and Player 2 has a combo of 29 because their last step received a PERFECT.

At the same time, there is a number at the bottom of the screen which represents their cumulative score.

Player 1's cumulative score is 16,125,310 and Player 2's cumulative score is 7,091,762.

Of course there are other criteria involved in game play which do not affect the score so I won't bother going through them here.

Procedure:
Originally I had played a song five times over with different combinations of each judgment and different combos. Then I set up five equations to see what each judgment was worth (each of which was expressed as an independent variable witch's multiplier was how many of that judgment I hit during a song) After having done this to almost 30 songs (each played five times) , I noticed that none of the information which I had drawn was consistent so the combo must be a factor towards the total score in some way.

So I thought that maybe paying attention to the value of each step would reveal how the combo plays a part in the scoring system, ten steps at a time.

Step #: The number of the step in the ten step sequence
Judgment: What judgment is awarded to the step
Score: The score awarded to the step
Total score: Sum of all scores including the last step
Combo: Combo including the last step

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 PERFECT 1,110 1,110 1 2 PERFECT 1,443 2,553 2 3 PERFECT 1,776 4,329 3 4 PERFECT 2,109 6,438 4 5 PERFECT 2,442 8,880 5 6 PERFECT 2,775 11,655 6 7 PERFECT 3,108 14,763 7 8 PERFECT 3,441 18,204 8 9 PERFECT 3,774 21,978 9 10 PERFECT 4,107 26,085 10

So the value of the next PERFECT step is 333 more than the last. (Let s = step)

s₂ = s₁ + 333

However that does not satisfy the first step (as 1,110 =/= 0 + 333)

1,110 - 333 = 777

So you could say:

( s * 333 ) + 777 = PERFECT score of step number s

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 GREAT 888 888 1 2 GREAT 1,221 2,109 2 3 GREAT 1,554 3,663 3 4 GREAT 1,887 5,550 4 5 GREAT 2,220 7,770 5 6 GREAT 2,553 10,323 6 7 GREAT 2,886 13,209 7 8 GREAT 3,219 16,428 8 9 GREAT 3,552 19,980 9 10 GREAT 8,885 28,865 10

Once again, the value of the next GREAT step is 333 more than the last. (Let s = step)

s₂ = s₁ + 333

However that does not satisfy the first step (as 888 =/= 0 + 333)

888 - 333 = 555

So you could say:

( s * 333 ) + 555 = GREAT score of step number s

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 GOOD/Boo/Miss... 0 0 0 2 GOOD/Boo/Miss... 0 0 0 3 GOOD/Boo/Miss... 0 0 0 4 GOOD/Boo/Miss... 0 0 0 5 GOOD/Boo/Miss... 0 0 0 6 GOOD/Boo/Miss... 0 0 0 7 GOOD/Boo/Miss... 0 0 0 8 GOOD/Boo/Miss... 0 0 0 9 GOOD/Boo/Miss... 0 0 0 10 GOOD/Boo/Miss... 0 0 0

Any combination of GOOD/Boo/Miss... score = 0

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 PERFECT 1,110 1,110 1 2 PERFECT 1,443 2,553 2 3 PERFECT 1,776 4,329 3 4 PERFECT 2,109 6,438 4 5 PERFECT 2,442 8,880 5 6 GREAT 2,553 11,433 6 7 GREAT 2,886 14,319 7 8 GREAT 3,219 17,538 8 9 GREAT 3,552 21,090 9 10 GREAT 8,885 29,975 10

Until step 6, things are identical to the all PERFECT scenario. After which, the 6th step is worth only 111 more than the 5th step but than the 333 difference resumes because all steps are then hit GREAT.

But let's just substitute in some numbers from the established PERFECT and GREAT equations.

( 1 * 333 ) + 777 + ( 2 * 333 ) + 777 + ( 3 * 333 ) + 777 + ( 4 * 333 ) + 777 + ( 5 * 333 ) + 777 + ( 6 * 333 ) + 555 + ( 7 * 333 ) + 555 + ( 8 * 333 ) + 555 + ( 9 * 333 ) + 555 + ( 10 * 333 ) + 555 = 29,975

This could also be expressed as:

333 ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 ) + ( 5 * 555 ) + ( 5 * 777 )

Or with variables:

333 ( Sum of number of steps ) + ( Number of GREAT * 555 ) + ( Number of PERFECT * 777 )

From these equations, I can conclude that it does not matter which judgment you earn for which step. You could earn 5 GREAT and then 5 PERFECT or each alternating and you would still get a score of 29,975.

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 PERFECT 1,110 1,110 1 2 PERFECT 1,443 2,553 2 3 PERFECT 1,776 4,329 3 4 PERFECT 2,109 6,438 4 5 PERFECT 2,442 8,880 5 6 Miss... 0 8,880 0 7 Miss... 0 8,880 0 8 Miss... 0 8,880 0 9 Miss... 0 8,880 0 10 Miss... 0 8,880 0

In this case, 5 steps were hit as PERFECT and 5 were hit as Miss... (Note, GOOD or Boo would also work in this case because they don't add to your total score either) You could express this as:

( 1 * 333 ) + 777 + ( 2 * 333 ) + 777 + ( 3 * 333 ) + 777 + ( 4 * 333 ) + 777 + ( 5 * 333 ) + 777 + 0 + 0 + 0 + 0 + 0 = 8,880

But will this work the same way as ten PERFECT and GREAT hits in the way that so long as there are always 5 of each, you will always get a score of 8,880?

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 PERFECT 1,110 1,110 1 2 Miss... 0 1,110 0 3 PERFECT 1,110 2,220 1 4 Miss... 0 2,220 0 5 PERFECT 1,110 3,330 1 6 Miss... 0 3,330 0 7 PERFECT 1,110 4,440 1 8 Miss... 0 4,440 0 9 PERFECT 1,110 5,550 1 10 Miss... 0 5,550 0

Clearly not. But from testing this, it is evident that this s that has up until now been used to indicate the step number in the sequence, it really the player's combo. So if the combo is "broken" by a GOOD, Boo or Miss... then s will reset back to 0. So more accurately, the previous sequence would go like this:

( 1 * 333 ) + 777 + ( 2 * 333 ) + 777 + ( 3 * 333 ) + 777 + ( 4 * 333 ) + 777 + ( 5 * 333 ) + 777 + ( 0 * 333 ) + ( 0 * 333 ) + ( 0 * 333 ) + ( 0 * 333 ) + ( 0 * 333 ) = 8,880

And this sequence would be:

( 1 * 333 ) + 777 + ( 0 * 333 ) + ( 1 * 333 ) + 777 + ( 0 * 333 ) + ( 1 * 333 ) + 777 + ( 0 * 333 ) + ( 1 * 333 ) + 777 + ( 0 * 333 ) + ( 1 * 333 ) + 777 + ( 0 * 333 ) = 5,550

Or as:

5 ( 333 * 1 ) + ( 5 * 777 ) = 5,550

This is because the combo of 1 is reached 5 times. 5 hits were PERFECT.

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 PERFECT 1,110 1,110 1 2 PERFECT 1,443 2,553 2 3 Miss... 0 2,553 0 4 PERFECT 1,110 3,663 1 5 PERFECT 1,443 5,106 2 6 PERFECT 1,776 6,882 3 7 Miss... 0 6,882 0 8 PERFECT 1,110 7,992 1 9 PERFECT 1,443 9,435 2 10 Miss... 0 9,435 0

Now for another case similar to the last.

( 333 * 1 ) + 777 + ( 333 * 2 ) + 777 + ( 333 * 0 ) + ( 333 * 1 ) + 777 + ( 333 * 2 ) + 777 + ( 333 * 3 ) + 777 + ( 333 * 0 ) + ( 333 * 1 ) + 777 + ( 333 * 2 ) + 777 + ( 333 * 0 ) = 9,435

Or:

3 ( 333 * 1 + 2 ) + ( 333 * 3 ) + ( 7 * 777 ) = 9,435

This is because the combo of 1 and 2 are both reached three times. The combo of 3 is reached once. 7 hits were PERFECT.

Now for a scenario with all the possible judgments.

Step # Judgment Score Total score Combo 0 N/A N/A 0 0 1 PERFECT 1,110 1,110 1 2 Boo 0 1,110 0 3 GREAT 888 1,998 1 4 GREAT 1,221 3,219 2 5 PERFECT 1,776 4,995 3 6 Miss... 0 4,995 0 7 Miss... 0 4,995 0 8 GREAT 888 5,883 1 9 GOOD 0 5,883 0 10 GREAT 888 6,771 1

4 ( 333 * 1 ) + ( 333 * 2 + 3 ) + ( 2 * 777 ) + ( 4 * 555 ) = 6,771

This is because the combo of 1 is reached four times. The combo of 2 and 3 are both reached once. 2 hits were PERFECT. 4 hits were GREAT.

Now that we know how the scoring system works, it's time to tackle the idea of this project: Would it be possible for two players to get the same number of each judgment but for one player to get a higher score? Or in other words, does combo really matter?

So let's say that both players get 159 steps PERFECT, 40 steps GREAT and one step Miss.... Player 1 gets a peak combo of 199 and Player 2 gets a peak combo of 100.

Player 1's score:

( 333 * 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 + 109 + 110 + 111 + 112 + 113 + 114 + 115 + 116 + 117 + 118 + 119 + 120 + 121 + 122 + 123 + 124 + 125 + 126 + 127 + 128 + 129 + 130 + 131 + 132 + 133 + 134 + 135 + 136 + 137 + 138 + 139 + 140 + 141 + 142 + 143 + 144 + 145 + 146 + 147 + 148 + 149 + 150 + 151 + 152 + 153 + 154 + 155 + 156 + 157 + 158 + 159 + 160 + 161 + 162 + 163 + 164 + 165 + 166 + 167 + 168 + 169 + 170 + 171 + 172 + 173 + 174 + 175 + 176 + 177 + 178 + 179 + 180 + 181 + 182 + 183 + 184 + 185 + 186 + 187 + 188 + 189 + 190 + 191 + 192 + 193 + 194 + 195 + 196 + 197 + 198 + 199 ) + ( 159 * 777 ) + ( 40 * 555 ) = 5,124,093

Player 2's score:

( 333 * 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 + 100 ) + ( 333 * 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 ) + ( 159 * 777 ) + ( 40 * 555 ) = 3,542,343

And it is this which is the flaw of the earlier DDR scoring system. Although they hit with identical accuracy, Player 1 annihilates Player 2 by a score of 1,581,750 points!

So what would be a more "fair" system? Because I have arranged the equations in a way which separates the "combo score" from the "judgment score", the " combo score" could be easily removed and both player's scores would look like this:

( 159 * 777 ) + ( 40 * 555 ) = 145,743

And thus, all the fun has been taken out of DDR forever.