Challenge Scoring Observations

[Epigenetic]

I had the idea for trying to unravel the mysteries of challenge score for several months. While I do not have a lot of answers, I did find out a few things. More answers will come after I am able to run a lot more tests. I saw that @alcaselzer posted something 3 days ago but I didn’t even see it until now…he or she is welcome to use my data.

Before I start, none of this is set in stone; I may have made errors in my observations. In addition, SG changes scoring in the challenges so exact formulas may break down over time. However, the concepts will endure.

I based my data on stage one of Rare (Beginner) but I did see the scores of some of the other stages. I am breaking this information up into three posts.

The enemy score total is not quite the sum of the monster hit points but is in the ballpark. It turns out that the health score is 80% of the enemy score. For the Rare stages, they are as follows (note that I only filled in what I observed; the others are easy to fill in):

Rare:
Stage____Enemy Score__Health Score__Stage Multiplier
1________5500_________4400________1.0000
2________6160_________4928________1.1200
3________6820_________5456________1.2400
4________7480_____________________1.3600
5________8140_____________________1.4800
6________9600_________7680________1.7454
7_______10320_____________________1.8764
8_______11040_____________________2.0073
9_______11760_____________________2.1382
10______15600_____________________2.8364

The intermediate and advanced are 1.5x and 2x the rare stage. I have informally looked at monster hit points before and I assume the other stages are the same but I did not verify this.

Stage____________Enemy Score__Health Score__Challenge Multiplier
Rare Stage 1________5500_______4400_________1.0
Epic Stage 1.________8250_______6600_________1.5
Legendary Stage 1.__11000_______8800_________2.0

The method for calculating these parts of the score is straightforward. The enemy score is like a constant, you get it for beating the stage. The health score is the fraction of your total hit points on your team and multiplies it by the perfect health score. For example, if your heroes all have 1000 hit points, and two are dead and three healed, your fraction of team points is 3000/5000. Let’s suppose you are on epic stage 3. The formula would be:

Your Health Score = Ideal Health Score of Rare Stage 1 x Fraction of Team Hit Points x Stage Multiplier x Challenge Level Multiplier

4400 x (1500/2500) x 1.24 x 1.5 = 4910 of a possible 8184 for the stage

I have not tested this extensively so there may be a small discrepancy for rounding and other factors. I do not think there is a penalty for dying although there is a continue penalty that I did not include in this analysis.

[Epigenetic]

The time score has an inverse relationship to the duration of the stage. All of this is based on stage 1 Rare. The data were surprisingly difficult to collect because I read some of the health and other scores instead of time scores, and since I was not doing videos at that point, scores could be lost.

The bottom line is that the relationship

Time score = A / (B + Stage Time in seconds) fits VERY well.

You can see the graph at the bottom and ignore the rest of the discussion. I tried one or two other stages, and it looks like stages 2-5 may use the same formula and multiply by the stage multiplier but that the remaining stages use different parameters, presumably due to the increased number of bosses. This would be easy to verify in future challenges. There are also some weird discrepancies, where a very slightly higher time can give a very slightly higher score but these may be due to the fact that the display timer may be slightly different than what the game uses.

Here is the raw data from stage 1 Rare (sorry about the formatting but I can’t get tabs to work)

Time (sec)___Time score___Predicted
49.2________10312_______10379
58.6_________9735________9665
73__________8695________8743
79.9_________8448________8361
88.7_________7870________7920
98.6_________7507________7476
118.3________6732________6727
152.1________5742________5739
176.7________5197________5185
193.2________4867________4869
211.9________4554________4555
221.5________4405________4409
232_________4224________4250
622.8________1864________1885

Here is the Mathematica code that I used. The program makes it easy to fit a model. The numbers may not look perfect, but if you look at the plot, biological experiments are never that clean

It should not be difficult to reproduce this for other stages. If you take a long time to do a stage, then a few extra seconds to heal yourself is no big deal. However, if you are doing this stage in well under a minute, every extra second counts. If you could do the stage in zero time, the model predicts you will score about 16,920 for time score. Overall, none of this is surprising and what most of us would have predicted based on experience.

[Epigenetic]

This post about match scores will be quite short. The scores will not be straightforward. I will give some extremely preliminary observations:

1. Most of my match scores on Rare stage 1 are 6000-9000

2. I was able to complete the stage (barely) by using only axes, arrows, and two sets of mana potions with all of the specials from Ulmer, Brienne, etc. I did not move one tile. The match score was zero!

3. I did the stage with only one match, a three tile normal one. My match score was 1848.

4. I did the stage with only one match making a four tile combo. The match score was 4125.

5. I did the stage with only one match, a combo with a three tile match leading to another three tile match. The match score was 6435.

6. I did a stage with three matches of three and one double match with two three item combos formed simultaneously (the two matches were in one horizontal line). The match score was 6385

7. A lot of stages with multiple matches (at least a dozen in some cases) had scores of 6000-7000. Ones where I tried to make diamonds were still 8000-9000

To me, this means that the match score may depend on some sort of average value of the matches and another value for the number of matches made, but it is not a direct multiplication of the values. A lot more tests will be done, similar to the ones I described with one action taken. At least we know that specials and items don’t give you any combo score, as expected. Also, if you do a stage very quickly and have a few matches, your combo score should not be affected too negatively.

I will post more after I look at the remainder of the videos, but don’t expect miracles. My job is super busy so it may be a month or two before I get to it, and even then I will need more videos to make any reasonable conclusions.

[alcaselzer]

Thanks dude for tagging me!
Your analysis is very well posed and the preliminary results quite interesting. As soon as I have time to look at the (few) videos I made yesterday, I’ll share the results with you if you are interested, mainly to add to your time score tabs and stage multipliers (only recorded rare stages to avoid at first discrepancies between challenges).
I find very interesting the two cases in which you scored better with a single 2xcombo than with the combo plus a couple of matches, which suggests some kind of average formula as you pointed out.
Thank you again dude for the job and for the sharing, I’ll keep in touch

[Eddard]

This information is great! Thank you for posting! It seems like this information could be used to determine a possible high score for each level and give you a range to shoot for instead of just “doing your best”. Lol @ “zero time”. How does he do it?!?

[Epigenetic]

@alcaselzer

I’ll share the results with you if you are interested

Sure alcaseltzer, I can use all the help I can get. During the next challenge we can see if the scores and formulas match. I need to think of some good experiments, but all of them use a lot of items. In beta, I ran out of axes and bombs after hitting mega titans so that didn’t help much.

[alcaselzer]

Sorry, it took me a while to have a look at my few (8) videos (all in rare challenge). I just recorded some of the attacks I had to do, did not perform specific attacks following a smart protocol for the sake of a best investigation, so don’t try to find a scheme in what I did.
By the way, for what enemy and health scores are concerned I can only add that:

1. I recorded a full health win on stage 8 and the score is 8832 as predicted;
2. I think that the stage multipliers work more or less like follows (does it really matter? don’t think so, but maybe it can help digging the developers minds):

• stages 1-5 (1 boss) → stage1*(1+0.12(stage#-1)); example stage 3 → 5500*(1+0.12*(3-1)) = 6820
• stages 6-9 (2 bosses) → stage6*(1+0.075*(stage#-6)); example stage 9 → 9600*(1+(9-6)*0.075) = 11760
• stage 10 (3 bosses) → stage6*(1+0.625) (nice coeff. imo); example stage 10 → 9600*(1+0.625) = 15600

3. I found that, as you said, the health score matches quite well 80% of the enemy score times the fraction of health left in the end. Example: heroes with 675 734 622 859 725 health were left with 493 713 611 595 725 respectively on stage 8. The predicted health score would be 96001.150.8*(493+713+611+595+725)/(675+734+622+859+725)=7664, while the actual score was 7661. Actually the formula seems to overestimate unless the match finishes with all heroes in full health, and the less health for my heroes, the greater the gap (more or less) between real score and predicted one, so maybe there is one missing factor. Here the complete data:
   Stage__Health Fraction__Health Score__Predicted Score
   6______0.93831________7200________7206.24
   6______0.88188________6768________6772.85
   6______0.99253________7622________7622.85
   7______0.86113________7100________7109.52
   8______0.86777________7661________7664.17
   8______1.00000________8832________8832
   9______0.94965________8925________8934.35
   10_____0.97538________12168_______12172.75

[alcaselzer]

Regarding time scores I can only share my data, but these figures alone are hardly enough to try a fit:

1. on stage 6 I have 3 pairs
   TIME__TIME SCORE
   92.1___12844
   106.1__11750
   113.6__11376
   estimated A and B for formula A/(t+B): A = 2.092298e6, B = 71.03
2. on stage 8 I have only 2 pairs
   TIME__TIME SCORE
   130.7__11923
   150.0__10863
   estimated A and B for formula A/(t+B): A = 2.358233e6, B = 67.09
3. Lone pairs on stages 7, 9 and 10:
   STAGE__TIME__TIME SCORE
   7_______93.1___13653
   9_______133.3__12559
   10______199.8__13525

Since in your data the formula was not perfectly fitting I don’t really trust this fits of mine with so few samples, the error in A and B estimation can be really large.
By the way, assuming they are reliable, I can’t tell if there’s a formula for A and B computation or if they are just values drawn from a look-up-table. It just seems that A increases with the stage multiplier while B decreases, but that is that, before trying further guesses I’d like to collect more data in next events.

[Epigenetic]

Thanks! I think the health scores are rounded. Also, the data for my A and B values is skewed with too many points in the middle of the curve, which I’m sure throws off the exact fit. I don’t think it is worth correcting more because the concept is the important thing.

I have another bit of news from beta and you have something else to digest

After that I am semi-retiring from more experiments in this area. It was fun figuring it out, but I want to spend my time getting good scores rather than figuring out all of the coefficients. If anyone wants to do that, be my guest!

[Epigenetic]

I am pretty sure I have broken the match score algorithm. This is all with Stage 1 of rare. I know stage 2 uses the multiplier and I suspect the other stages and difficulty levels follow a similar multiplier to the health and monster numbers but I have not tested it yet.

The gory details are in the next post, but the bottom line is:

The match score takes the average number of tiles in each move over the entire stage run, and then uses a formula to figure out the score. The match score increases while your average increases, but it is a law of diminishing returns.

A couple of examples will suffice and for those who are interested I put the rest in another post.

When you do a move, you get rid of at least three tiles (There is a rare exception of hitting a diamond with zero or one matching tile but if you are bothered by this you can figure out why it doesn’t matter ) . Here are some examples:

Match one set of three tiles = 3 tiles (duh)

Make one diamond with 5 tiles = 5 tiles (some diamonds are made with more tiles)

Make a combo with a three tile match followed by another three tile match = 6 tiles

Set off an existing diamond with 9 additional same color tiles = 10 tiles

Large combinations can have 30 or more tiles

All of these are as a result of one move. Take all your moves, and the number of tiles, and average them.

Match average = Number of tiles removed / Number of moves

That’s it! Here are some averages and the match score you generate for rare stage 1:

3 = 1848
3.5 = 3168
4 = 4125
4.33 = 4675
5 = 5500
5.5 = 6028
6 = 6435
7 = 7095
7.5 = 7348
8 = 7568
9 = 7953
13.5 = 8965
15 = 9185

Now we know why match scores for this stage are often 6000 to 8000, because most people will have a few large combos. If you use a bunch of items and make only match 3s, your match score will be low. The bench clearing stages where multiple diamonds go off will make your score exceed 9000, but there are diminishing returns.

Remember that these match scores will go up with stage number and with difficulty level.

I will let you draw your own conclusions, but briefly as long as you make diamonds and clear lots of space, your match score will be high. Clearing one level with items usually won’t affect the score since it is an average.

[Epigenetic]

Here are the details of how I figured this out. I did the level with very few matches and could isolate various aspects. There were other examples but I put in the highlights

1. The match score is an average

One three tile match gives a match score of 1848. This is the same score as two individual three tile matches and seven three tile matches

One four tile match is a score of 4125, and this is the same as two individual four tile matches.

2. There is no bonus for making diamonds or bombs (although you will benefit from these later)

One five tile match + one three tile match = score 4125. This is eight tiles, and is the same score as two four tile matches. That shows the diamond is the same as two bombs, and I have other examples in which a bomb and a diamond don’t have any score bonus vs regular three tile matches, so both “bonuses” are zero.

3. There is no bonus per se from making a big or small combo other than adding together the tiles that are matched. This is different than scoring in other areas where big combos give a progressively larger bonus.

One three tile match and then one combo with five three tile matches in a row = 3 + 15 = 18 tiles which is 9 tiles per move. The score was 7953. That is the same score as one combo with three 3 tile matches = 9 tiles in one move.

I fit the data to a model:

Match score = C – (A/X) where X = average number of tiles.

(note that this is edited from the initial post)

The fit is quite good. Note that one data point from my previous post last month was incorrect so I threw out that point. The x axis starts close to 3, not at 0. Finally, if you have no matches at all, the formula doesn’t apply but your match score is zero as you would expect.

[alcaselzer]

These results are awesome, really!
I don’t know how you chose the model to fit, do you have a good eye on guessing non linear equations or did you use genetic programming?
Either cases that’s really impressive, congrats!

Many thanks for sharing

[Epigenetic]

I edited the post to make the formula more consistent and simpler. Really, the first formula for time can almost be rearranged to A + B/X, and the second for match to A - B/X. That would be symmetrical but it doesn’t quite work out.

As to how it was figured out, things that have some sort of asymptote are usually division or exponential or logarithmic. I haven’t programmed in decades, but when we used to use Z80s and Motorola 6502s, divisions were faster than exponent calculations. I tried both and the division worked well.

I used to be quite good in math and programming but if you think I am smart then (1) talk to my wife, and (2) consider this. I told my father in 1984 that I thought I could get a job at a couple of companies called Apple and Microsoft. He told me “there is no money in computers” and my career took a completely different path. Had I been successful at one of those firms, I could own SG games. As it is, I have a rewarding job but I have be content with paying for Guin! I should have a statue of myself in my house on permanent display with a massive dunce cap on, and a tiny part of the penance is to help figure out this formula.

[Ellilea]

Event is upon us with the pushing day tomorrow so this could be a nice read for the folk who can strive to better their scores!

[BarryWuzHere]

Thanks for this, it is great info!

I would be curious to see how “winning” or at least “competitive” scores in the challenge are distributed–how many points come from the time component, the match component, and the health component.

I’ve also got a very interesting question about the health score: I use Delilah heavily, and like to go into the boss level with three minions in front of all my heroes, putting them ~450HP over their max. When I finished stage 10 (Attack items to blast the bosses nearly dead), all my heroes were over max including them. I’m wondering if that gave me bonus points above “full health”?

I have spotty screenshots; this might be enough data to make a guess on whether minions bonus HP go into a health score or not…enjoy!

List of my scores using Delilah

Stage Enemy Time Match Health
Legendary Stage 2 11,660 6,051 16,440 9,234
Legendary Stage 3 12,320 5,913 17,740 9,856
Legendary Stage 4 12,980 7,281 19,158 10,384
Legendary Stage 6 15,600 7,581 19,812 12,152
Legendary Stage 7 16,320 6,168 24,414 12,794
Legendary Stage 8 17,040 7,105 22,322 13,086
Legendary Stage 9 17,760 5,860 24,260 9,199 (one hero down, I think)
Legendary Stage 10 23,100 11,919 31,554 18,480 (Minions put me over full HP for sure on this stage, I think it was seldom the case otherwise)

Edit: I just re-found this, and I’m now 99% sure that minions don’t help except that they make it easier to keep your regular health full.

[Ebonest]

This is one of the great threads in the forum, so i’ll just bump it for everyone to see.