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Given the lack of answers, I'm starting an experiment today - a savefile with global drop chance and a savefile without. The monsters killed by offline dungeon can be used to measure the drops (after enough time to get relevant data) - both kills and drops can be reliably counted.
If the global drop chance does nothing, it's safe to assume that drop chance modifiers aren't a thing for glowing mana shards.
https://steamcommunity.com/sharedfiles/filedetails/?id=2761906569
Although it does not tell you the drop chance of Glowing Mana Shards, it does tell you the modifiers on their drop chance. It does not look like global drop chance has any effect.
Btw, do yoy ever do dark rituals (that resource drop from class)?
I did hourly dark rituals for a while, but I put that on pause to collect seeds (and exp) and let the multipliers from the Tamer and Warrior routines build up. The dark rituals were giving diminishing returns and it is nice to take a bit of a break.
I will need to spend a lot of Sin to increase Inspiration gain to make Dark Rituals more worthwhile.
I thought that perhaps a game update increased the drop chance, but maybe it is because of the global drop chance from the Tamer routines building up?
The experience can benefit from the DRs though - Valuable Experience seems to have a polynomial scaling with level, so its effect still goes up at high levels, not a quick drop to nothing. I'm currently building an exp run on Demon King (1.63M Valuable Experience, +6.39k% exp now) - it already makes a difference.
The seed collection doesn't exactly depend on tamer routines, bosses are naturally better than normal mobs; no mimics from them though. If anything, it's the envy routine that matters, especially with dragon village summons.
Edit:
Checked the test saves. The numbers are still highly inconsistent (600 to 2000 monsters per shard with the same setup). This should get better with more time. For now it seems to be roughly 1k kills per 1 glowing mana shard if global drop chance does nothing (not enough data to determine).
You know what, I'm also interested in this, so I'll keep track of my average Glowing Mana Shards per kill for a while and see if it is any different from what you are getting.
EDIT: So far I got 8 shards in about 8000 kills, so your guess of 1000 being the average seems pretty accurate.
EDIT2: 52 shards in 37430 kills, which averages to 720 kills per shard.
EDIT3: 98 shards in 73600 kills, which averages to 751 kills per shard.
EDIT4: 182 shards in 142000 kills, which averages to 780 kills per shard
Some more numbers:
Equipment global * 1.42, Rocky Hill, 584148 kills, 763 shards, 766 kills/shard
Equipment global * 1.42, Holy City, 584046 kills, 811 shards, 720 kills/shard
No equipment global, Rocky Hill, 585891 kills, 770 shards, 761 kills/shard
The global drop chance most likely does nothing. Monster type probably also doesn't matter.
I did 5 loads of the save and summed both kills and shards. The shard drops from 1 batch of ~117k kills ranged from 53 to 260; there might be a lot of randomness involved.
To find out the kills, I multiplied the kills/minute with the amount of minutes that has passed since I started keeping track.
I now tried your method by loading an old save 48 times.
In total this gave 7189 shards in 4.6 million kills, with an average of about 643 kills per shard.
It seems that indeed a much larger sample size is needed for accurate results.