Forewords
Big thanks to Tin Man for this topic in which, collectively, huge advances were made on the problem of "how much experience is needed to reach a level?".
Also, to the one who found the formula for high champion level. The post has been lost due to the October post-hack rollback, but I think it was from Argon in a follow-up to his previous post.
Summary of Findings
I have done a lot of experiments around all the (tentative) formulae we came up with, and until proven otherwise the one I trust to be accurate for computing the amount of XP needed to reach the next level are :
(In the last formula the ⌊…⌋ brackets denotes rounding down.)
Examples: According to the relevant formula, 12 650 567 XP points separate CL 1122 from CL 1121, and 12 740 559 XP points separate CL 1126 from 1125. (I have strong evidence that these amounts are exact.)
From these come the formulae to compute the total amount of XP needed to reach any level:
(NB. The last formula is an approximation, we don't know whether the XP increments are truncated internally. If that is the case the formula is a bit inaccurate, with an overestimation growing almost linearly by close to half a point for each level above 700.)
Example: According to the relevant formula, a total of 4 999 040 champion XP points are needed to reach CL 73. (Again, I have strong evidence that this amount is exact.)
About the Inverse Problem
Given an amount of accumulated XP, what is the corresponding hero or champion level? There are closed formulae for this, at least for all hero level and for champion level up to 699. But they are rather involved (lots of cubic and square roots) and of limited use, so i won't give them here. You can use (as I did) a computer algebra system to retrieve them.
The simple way here is the better, if only because it work at and above CL 700: just tabulate all the amounts for all hero level and for champion level up to e.g. 2400 (that's 6 595 341 625 486, a little bit more than 62 time the amount needed to reach CL 2100) and do lookups.
About Experience Merging
After some experiments (with surprising results!) I do believe that the game maintain two experience counters, at least for an event toon: one for the XP gained while the toon wasn't yet at level 70, one for the XP gained after that. Note that I have some evidence that all the XP gained before reaching level 70 is kept in the first counter (the "hero" experience counter), so it may go above the amount needed to reach level 70 (e.g. because the last kill overreached that amount).
When the event is ended with no slot for the event hero, both experience counters are added to the account "champion" experience counter. Otherwise, only the event toon "champion" experience counter is added to the account one.
If this is true I think it's a nice touch: no effort put into the event toon is lost, even if it wasn't enough to reach level 70.
Post Scriptum
This last bit is confirmed here, thanks to Nhat for the link.
Post Scriptum Bis
I have made a spreadsheet for XP computations, available in my Dropbox (link in my signature).
Edited by LodWig; 01-06-2020, 03:11 PM. Reason: Changed the formula for CL above 700, with rouding down instead of rounding to nearest.
Big thanks to Tin Man for this topic in which, collectively, huge advances were made on the problem of "how much experience is needed to reach a level?".
Also, to the one who found the formula for high champion level. The post has been lost due to the October post-hack rollback, but I think it was from Argon in a follow-up to his previous post.
Summary of Findings
I have done a lot of experiments around all the (tentative) formulae we came up with, and until proven otherwise the one I trust to be accurate for computing the amount of XP needed to reach the next level are :
| Hero Level |
| Range of current level ℓ | XP needed to reach next level |
| 1 | 95 |
| 2 — 49 | ( 40 × ℓ + 150 ) × ℓ |
| 50 — 69 | ( 120 × ℓ + 750 ) × ℓ |
| Champion Level |
| Range of current level ℓ | XP needed to reach next level |
| 0 — 699 | ( 10 × ℓ + 30 ) × ℓ + 50 000 |
| 700 — ∞ | ( 10 × ℓ + 30 ) × ℓ + 50 000 + ⌊ 1.015 ( ℓ - 700 ) ⌋ |
(In the last formula the ⌊…⌋ brackets denotes rounding down.)
Examples: According to the relevant formula, 12 650 567 XP points separate CL 1122 from CL 1121, and 12 740 559 XP points separate CL 1126 from 1125. (I have strong evidence that these amounts are exact.)
From these come the formulae to compute the total amount of XP needed to reach any level:
| Hero Level |
| Range of level ℓ | Hero XP accumulated to reach this level |
| 2 — 50 | 5 × ( ( ( 8 × ℓ + 33 ) × ℓ - 41 ) × ℓ - 57 ) / 3 |
| 51 — 70 | 5 × ( ( ( 8 × ℓ + 63 ) × ℓ - 71 ) × ℓ - 793 819 ) |
| Champion Level |
| Range of level ℓ | Champion XP accumulated to reach this level |
| 1 — 700 | 10 × ℓ × ( ℓ × ( ℓ + 3 ) + 14 996 ) / 3 |
| 701 — ∞ | 10 × ℓ × ( ℓ × ( ℓ + 3 ) + 14 996 ) / 3 + 200 × ( 1.015 ( ℓ - 700 ) - 1 ) / 3 |
(NB. The last formula is an approximation, we don't know whether the XP increments are truncated internally. If that is the case the formula is a bit inaccurate, with an overestimation growing almost linearly by close to half a point for each level above 700.)
Example: According to the relevant formula, a total of 4 999 040 champion XP points are needed to reach CL 73. (Again, I have strong evidence that this amount is exact.)
About the Inverse Problem
Given an amount of accumulated XP, what is the corresponding hero or champion level? There are closed formulae for this, at least for all hero level and for champion level up to 699. But they are rather involved (lots of cubic and square roots) and of limited use, so i won't give them here. You can use (as I did) a computer algebra system to retrieve them.
The simple way here is the better, if only because it work at and above CL 700: just tabulate all the amounts for all hero level and for champion level up to e.g. 2400 (that's 6 595 341 625 486, a little bit more than 62 time the amount needed to reach CL 2100) and do lookups.
About Experience Merging
After some experiments (with surprising results!) I do believe that the game maintain two experience counters, at least for an event toon: one for the XP gained while the toon wasn't yet at level 70, one for the XP gained after that. Note that I have some evidence that all the XP gained before reaching level 70 is kept in the first counter (the "hero" experience counter), so it may go above the amount needed to reach level 70 (e.g. because the last kill overreached that amount).
When the event is ended with no slot for the event hero, both experience counters are added to the account "champion" experience counter. Otherwise, only the event toon "champion" experience counter is added to the account one.
If this is true I think it's a nice touch: no effort put into the event toon is lost, even if it wasn't enough to reach level 70.
Post Scriptum
This last bit is confirmed here, thanks to Nhat for the link.
Post Scriptum Bis
I have made a spreadsheet for XP computations, available in my Dropbox (link in my signature).
Edited by LodWig; 01-06-2020, 03:11 PM. Reason: Changed the formula for CL above 700, with rouding down instead of rounding to nearest.
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