RS Difficulty Calculations Method

Operation Portion; Sourced from 鼓瑟吹笙LB

Rolling Sky Difficulty Calculations

This page contains formulas of how to calculate part of the final difficulty.

There are 2 parts to the final value:

Operation
Memorisation

Only the operation formulas are revealed to the public. Thus, you are not able to calculate the final value. However, the values calculated here, could be a good gauge of how difficult the level is as compared to the other levels. The list of the operation values for part of the levels are listed in the image below.

From the left,

the first black value denotes the marks for basic way;

the first red value denotes the difficulty value for basic way (that would be added to memorizations formula)

the second black value denotes the marks for perfect way;

the second red value denotes the difficulty value for perfect way (that would be added to memorisation formula)

[These values are in Chinese, if you do translation you may roughly guess what these levels are]

The section below is the list for all the formulas. The version below is done on October 2019, however, there has not been much change since then, and the formulas are still applicable. Just that the scores for example, Happy's slime drop, are not known. As always, take it as a reference number.

For example, from the calculations, A Dream of 7 Year's Operation difficulty is 4.58, as compared to The 4th Anniversary 4.49 and Cube 4.50, so the predicted difficulty should be 6.9-7.0, making it the hardest level (theoretically) [By 星陨_STAR]

[RS Operation Difficulty Operation]

[Original Article: 鼓瑟吹笙LB]

The difficulty is derived from the sum of scores from multiple high-difficulty actions. I believe summing up the scores is a very reasonable way to calculate the "overall difficulty." On the other hand, calculating the "error rate" involves integrating the square/cube of the difficulty density, and assessing the "player's experience" takes into account the percentage of occurrences of specific operations. These are separate considerations and will not be discussed further in this article.

Now, I will provide the difficulty scores for each action obtained through algorithms and statistics.

Using the difficulty of a rotation jump as the benchmark:

Rotation Jump Difficulty = (a² + b²) / 2, where a and b are the number of grid spaces the player needs to move before and after the obstacle, respectively. (For example, the rotation jump difficulty of Fairy Tale 1st crown is 1, while the rotation jump difficulty of Poker 4th Diamonds is 2.5). Round to one decimal place. For Big Jump (the blue box), scale it proportionally to the length of the jump board before calculating.

Below are examples of difficult terrains (numbers will be used in the following explanations to replace names):

Single Grid Turn (1.5 points each): The basic unit of an S-turn, where the ball passes through the 2 objects vertically (one object in front, one object behind)
Diagonal (2 points each). A three-grid "horizontal turn" without obstacles is the same concept as a diagonal.
Single Obstacle 1X3 Horizontal Turn (3 points each): [City 1st Diamond, Cube 2nd Box, etc. also falls in this category] Similar terrains need separate calculations for the diagonal turn after the Cube's 5th Diamond horizontal turn. In the "horizontal turn derivation formula" at the end of this section, this terrain is calculated as 2.5 grids. Note that Cube's 5th Diamond is not this terrain, as its landing point is in the middle of a grid, and the horizontal turn space is a bit more than half normal, requiring the "horizontal turn derivation formula" for calculation.
Obstacle 1X3 Horizontal Turn & Mover 1X3 Horizontal Turn (5 points each): True three-grid horizontal turns (i.e., requiring traversing 3 grids, especially when the third or first grid has a jump board/collectible). Classified as "normal type" (Microchip 1st Crown, Illusion Wave 6th Diamond, etc.) and "obstacle after horizontal turn type" (many are found in Naval Battle and Pirate Ship). The former is 6 points each, the latter 8 points each.
1X4 Horizontal Turn (8 points each)
Pseudo Single Grid Turn (1 point per set): Where one of the obstacle grids is empty. A set refers to several consecutive same-direction terrains not forming an S-turn arrangement, like the ones after the World Cup roller formation. Equivalent terrains (like mover diagonals or jump pads extending one grid on the side of the path) are also 1 point each (can be cumulative!).
Landing Rotation Jump (3 points each): Where there's an air obstacle one grid before landing (e.g., Egypt 46%).
Half-speed Three-grid Turn (1.5 points each): Passing between obstacles spaced two grids apart, like the 1UP's entry to 3rd box. If one obstacle is empty, it counts as 1 point.
Single Jump Pad/Single Obstacle Diagonal (4 points each)
Diamond one grid before/after obstacle or jump pad one grid after obstacle (1.5 points each): Treated as a single grid turn. Items on the ground before/after obstacles count as 1 point each.
Z-shake (starts at 2 points per diagonal turn): Arithmetic sequence with a common difference of 2 (i.e., 2, 4, 6, 8...).
Varying Christmas Small Jump Pads (2 points per jump): If arranged diagonally, 2 points per jump. If staggered by one grid, 1 point per jump.
Mover Diagonal (1 point each): Different from point 6's "1 point per set", this one is cumulative.
Single Grid Turn on 2nd floor (1.5 points each): Although there are no obstacles, the error tolerance time is the same as a regular single grid turn due to no 卡位
Two-grid Horizontal Turn Collecting Floating Diamonds (2 points each)
Half-speed 1X4 Turn (2.5 points each): Like the 1UP's entry to 4th box operation.
Forced Swipes from American Football (1 point each)
1X3 Movement Collecting Floating Diamonds (5 points each): (e.g., World Cup 8th Diamond)

Appendix: Horizontal Turn Derivation Formula: Horizontal turn a / horizontal turn b ≈ [a horizontal grid count * b vertical grid count]² / [a vertical grid count * b horizontal grid count]². For a three-grid horizontal turn, use 2 for the horizontal grid count, and so on. Adjust the results slightly based on actual gameplay feel.

Additional Considerations:

Correlation of Operations (see supplementary section)
Halloween Night Portal (different positions): 8 points each. Operations within 8 grids after teleportation are doubled in score. (The doubling of post-teleportation operation applies to any forced movement terrain.)
Ball Speed Formula: 1.6 * (ball speed per second / 10)²
Old Version: Considering the ball speed issue, the normal speed is temporarily divided into 5 levels (1, 2, 3, 4, 5), plus slow down (-1.5 levels) and speed up (10.3 levels), making a total of 7 levels. At a certain ball speed, all scores will be multiplied by (0.7 + level/10)² (squared because not only does the hand speed requirement increase, but the error tolerance time also shortens), and some difficulty actions will be added as necessary.
If half of the relevant obstacles/tile of a high-precision operation unit is moving significantly (including vertically) (e.g., S-turn at 74% in Halloween Night), difficulty *1.5 (due to special hitbox of rollers, this rule does not apply to them"); if a rotation jump can only be operated within a certain range (e.g., if the player cannot swipe to the third lane when conducting a rotation jump to collect the 1st crown of Midnight Carnival), then the difficulty is multiplied by 1.5

4b. Dual ball mode operation *1.5.

Mid air High-Precision Operation *2 (e.g., the routes for Faded Remix 3rd Crown and some point in Ignite involve mid-air single grid turns")

5b. if two related obstacles/floor of a high-precision operation unit are moving significantly (e.g., neon roller at this one part, World Cup 1st Crown), then *2 (rollers are also not taken into account here)

5c. Reverse operation *2. Scores for operations beyond 5 grids 2 (e.g., the first outer rotation jump in Anniversary 4th Box is 10*2=20); score for operations during tilt *2; operations during the pushback mechanic from Midnight Carnival *2 (though it hasn't appeared yet).

Random obstacles/diamonds are 3 points each. Random jump boards are 2 points each if they are continuous (arranged in a continuous sequence of 13524 and spaced no more than 5% level progress. A counter example is the the random jump pads in the basic way of Spring Festival after several of them in perfect way are skipped), otherwise 3 points each.
Horizontal turn whose destination falls on the outermost lane, score *1.2 (e.g., Naval Battle 1st Crown).
In random player formations with the number of players ≤ 5, the difficulty of the random part = number of players - 2 (not less than 0). A formation with five players plus a rugby ball is 4 points each. Add the mathematical expectation of the difficulty terrain scores that may appear to this basis as the overall random difficulty. (Microchip random difficulty is 1.5, diagonal operations will inevitably appear, so the difficulty sum for 9 random floors is 49.5 points.)
Reverse operation & acceleration: Movement difficulty = minimum number of grids needed to move. During tilt operations, 1 point per grid in the difficult direction, 0.5 points per grid in the easy direction. Midnight Carnival style backward route 0.5 points per grid. (Additional operations receive additional points.)
If the difference in the number of left and right grids between two jump boards during acceleration is n≥2, then an extra (n²/2) points each. (e.g., the second acceleration in Street Basketball). The same applies to difficult directions during tilt operations.
For some composite terrains (e.g., roller in the deep space segment of the fourth anniversary, horizontal turn at the 76%-77% junction of the desert segment), if two terrains are combined (e.g., the former is a combination of terrain 1 and 8) but cannot be separated, and if the composite does not interfere with operations (like the above horizontal turn, same direction and does not require higher hand speed), take the maximum value of the two.
"No-obstacle single grid turn" (terrain consisting of two parallel adjacent diagonal turns): In normal routes of 3-star levels and above, it does not consider operational points. But if it has strong visual confusion or requires additional points in acceleration segments, it counts as 0.5 points per set. For 2-star levels and below, also consider it as 0.5 points per set.
For levels with operation difficulty scores < 20, apply the following correction: Assign 0.1 points to "long single grid turn" (passing through the diagonal 2 grids of 8 grids arranged in "目" shape/one grid obstacle and one grid empty terrain) originally 1.5 points each or 1 point each in the terrain calculation. (This item is to describe the contribution of obstacle density and movement volume to difficulty in low-star levels.)

I ALSO HAVE NO IDEA WHAT DOES SKILLED AND NOT SKILLED MEAN

Final Calculations:

Calculation for Players Not Skilled in Operations:
- Use the formula (score)1/3.5(\text{score})^{1/3.5}(score)1/3.5 to determine the operation star level.
- If the score is less than 10, divide the score by 5 instead.
Calculation for Players Skilled in Operations:
- Use the formula ln⁡(score)1.3\frac{\ln(\text{score})}{1.3}1.3ln(score) to determine the operation star level.
- If the score is less than 1, set the operation star level to 0.
Final Star Level:
- The final star level is obtained by adding the operation star level to the background star level.

Example Calculation

For a given score of 50:

For players not skilled in operations:
- Operation star level = (50)1/3.5≈2.57(50)^{1/3.5} \approx 2.57(50)1/3.5≈2.57
For players skilled in operations:
- Operation star level = ln⁡(50)1.3≈2.45\frac{\ln(50)}{1.3} \approx 2.451.3ln(50)≈2.45

Example Calculation for Low Scores

For a given score of 5:

For players not skilled in operations:
- Operation star level = 5/5=1.05 / 5 = 1.05/5=1.0
For players skilled in operations:
- Operation star level = ln⁡(5)1.3≈0.98\frac{\ln(5)}{1.3} \approx 0.981.3ln(5)≈0.98

Summary

By applying these formulas, you can calculate the operation star level tailored to different skill levels and then add it to the background star level to determine the final star level of the player.

Supplementary

1. Correlation of RS Operations

Classification of Relationships:
- Operations are classified as either "same direction" or "opposite direction."
- Only adjacent operations are considered, meaning no other operations are in between the two being evaluated.
Same Direction Operations:
- Case 1: If the operations are separated by only 1 grid (e.g., connected same-direction diagonal turns), there are no extra points, and similar single grid turns are not accumulated (as stated in the basic judgment).
- Case 2: If the operations are separated by 2 to 3 grids:
  - Extra points = (Score of operation A * Score of operation B) ÷ (Number of separating grids) ÷ (Vertical length of operation A)
  - This rule accounts for the difficulty in maintaining rhythm when continuous same-direction operations are separated by an interval requiring deceleration and a stop.
- Case 3: Operations separated by more than 3 grids are not considered.
Opposite Direction Operations:
- Case 1: If the operations are separated by 2 grids or less:
  - Extra points = (Score of operation A * Score of operation B) ÷ (Number of separating grids * 2) ÷ (Vertical length of operation A)
- Case 2: Operations separated by more than 2 grids are not considered.

2. Z-Turn Operations (Z字抖) Correlation Algorithm:

Z-turn operations follow the same algorithm as above.
When Z-turn operations are related to other operations, they are considered as "whole operations" for calculation, using only the score of the first diagonal turn of the Z-turn as the "operation B score." The subsequent diagonal turns are still arranged in the sequence of 4, 6, and 8 points respectively.
This specific rule primarily applies to complex scenarios like Poker 3rd Crown, due to the limited impact of the initial rotation jump on subsequent Z-turn operations.

3. Allocation of Extra Points:

Extra points, except for the first diagonal turn of the Z-turn, are added to the latter operation in the sequence.
The cumulative score is then used as the base score for the next operation's calculation.

Example Calculation:

Same Direction Example:

Case 1:

Operations A and B are separated by 1 grid.
No extra points added.

Case 2:

Operation A has a score of 3, and operation B has a score of 4.
They are separated by 2 grids, and the vertical length of operation A is 2 grids.
Extra points = (3 * 4) ÷ 2 ÷ 2 = 3 points.

Opposite Direction Example:

Case 1:

Operation A has a score of 4, and operation B has a score of 5.
They are separated by 2 grids, and the vertical length of operation A is 3 grids.
Extra points = (4 * 5) ÷ (2 * 2) ÷ 3 = 1.67 points.

By integrating these rules, a comprehensive and nuanced calculation of the final score can be achieved, accommodating both same and opposite direction operations with varying intervals.

Introduction by Y3G

Scores Image by 鼓瑟吹笙LB

Original Article by 鼓瑟吹笙LB

Translated by ChatGPT

Reviewed by Y3G & Petruso

Page updated

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