I’ve been playing Empires and Puzzles (a mobile match-3 RPG game, developed by Small Giant Games) for a few months and recently I joined a competitive alliance that fights 14-star titans on a daily basis.
With my limited hero pool, I have to think out-of-the-box to deal enough damage to help the alliance beat every single titan. I’ve seen people using titan shield to fight 14-star titans online, and the damage is comparable to what a decent all-5-star titan team + a lucky board (easily 60K per flag). But what’s the most effective way to use the costly titan shields? We’re going to find out with some experiments.
Assumptions
- Titan’s damage is calculated in the same way as hero damage:
Credits to MuaddibMcFly (full post)
2. Based on my experience, one would be able to use 3~4 titan shields in a single titan fight. For the experiments, we’re assuming we’ll use 4 shields per titan fight with about 10s left. That translate to some set up work and 16 titan round of titan actions.
3. For simplicity, we only analyze the normal attack damage. i.e. total damage would be:
damager_per_hit * (total_round - number_of_stunned_round)
where “total_round” means the round when your team has active titan shield on.
4. All heroes are level 1 and un-ascended. The 1-star hero used here is Fletcher due to lowest defense. All troops used are the 1 star, level 1, low-defense troops.
Experiments + Analysis
4 different line-ups:
- Empty + 3 * Fletcher + Empty.
- Empty + Muggy + 2 * Fletcher + Empty
- Gunnar / Kailani + Muggy + 2 * Fletcher + Empty
- Gunnar / Kailani + 3 * Fletcher + Muggy
Titan shields are always applied to the 3 heroes in the middle (the one in bold). Other than titan shields, the only item used is minor mana potion.
Defensive stats(with troops):
- Fletcher: 146
- Muggy: 223
- Fletcher Chameleon: 73
- Muggy Chameleon: 112
The goal is to find out which line-up / strategy is statistically the most efficient. Actual damage number may vary, and I’ll attach some video as reference
Line-up #1
Reference Video. Courtesy of Farmer Ben.
Line-up #1 is used to as a baseline. Let’s assume the total damage per hit for line-up #1 is 100.
Line-up #2
Reference Video. Courtesy of leugimrc.
Round #1, mana potion Muggy and cast the spell. First titan hit will kill one of your heroes and after round #1 you’ll have 1 * chameleon with 2 *heroes. Note that there is enough time for doing this and 16 rounds of titan actions.
For simplicity, the analysis is done based on this line-up after round 1. You can optionally spend another round to try converting another chameleon but the odds gets things complicated.
1/3 of the chance, you’ll have 2 * Fletcher and 1 * Muggy Chameleon. A single titan hit for this line up would be (according to the damage calculation):
2/3 * 1 + 1/3 * (146/112)^1.35 = 1.1434 times compared to line-up #1
2/3 of the chance, you’ll have 1 * Fletcher, 1 * Muggy and 1 * Fletcher Chameleon. A single titan hit for this line up would be (according to the damage calculation):
1/3 * 1+ 1/3 * (146/223)^1.35 + 1/3 * (146/73)^1.35 = 1.3712 times compared to line-up #1
So in total, the output damage per hit would be:
1/3 * 1.1434 + 2/3 * 1.3712 = 1.2953 times compared to line-up #1
If total damage per hit for line-up#1 is 100, line-up#2 would be 129.53
Line-up #3
Reference Video. Courtesy of Hikmet.
Round #1, mana potion Muggy, Gunnar / Kailani and cast the spell. First titan hit will kill all your team and after round #1 you’ll have 1 *Muggy chameleon and 2 * Fletcher chameleon and 1 * Gunnar / Kailani chameleon. The defensive buff from Gunnar / Kailani is gone after respawning as a Chameleon. Note that there is enough time for doing this and 16 rounds of titan actions.
If titan hits the shielded heroes, damage per hit would be:
2/3 * (146/73)^1.35 + 1/3 * (146/112)^1.35 = 2.1762 times compared to line-up #1
Note that Gunnar / Kailani chameleon is not covered by the titan shields at all, during one of the 16 rounds, titan may hit Gunnar / Kailani chameleon and not reflecting the damage. The odds of that happening is:
1 - (1 - 0.25)¹⁶ = 0.989977
Taking that into consideration, the 98.9977% of the times, this lineup will reflect 15 times instead of 16 times. You may argue that Titan AOE may do the favor for you, so keep in mind in reality the expected output damage would be higher than the following number:
2.1762 * (98.9977% *15 / 16 + 1.0023% * 16/16) = 2.0416 times compared to line-up #1
If total damage per hit for line-up#1 is 100, line-up#3 would be 204.16
Line-up #4
Reference Video. Courtesy of TPK’s Channel.
Round #1, mana potion Muggy, Gunnar / Kailani and cast the spell. First titan hit will possibly kill all your team except for Gunnar / Kailani. After round #1 you’ll have 1 *Muggy chameleon, 3 * Fletcher chameleon and 1 * Gunnar / Kailani chameleon (60%) or 1 * Gunnar / Kailani(40%).
The defensive buff from Kailani is gone after respawning as a Chameleon. Note that there is enough time for doing this and 16 rounds of titan actions.
If titan hits the shielded heroes, damage per hit would be:
(146/73)^1.35 = 2.5491 times compared to line-up #1
Similarly, the 2 uncovered heroes (Gunnar/Kailani and Muggy chameleon) will likely take 1 or 2 or 3 hits out of the 16 rounds of titan actions.
Gunnar / Kailani will take 1 hit to be killed, then possibly 1 more hit if he/she is killed under the chameleon buff, where Muggy chameleon will be killed with 1 hit. In total 16 rounds, the odds of landing 13, 14, 15 or 16 reflections is non-trivial to calculate. But over 90% you will land only 13 hits (if Gunnar / Kailani is not killed on the first round) or 14 hits (if Gunnar / Kailani is killed on the first round).
The odds of reflecting 16 hits is 0.6¹⁶ = 0.02821% and 15 hits is 1.449%. The remaining 98.523% consists of 14 hits and 13 hits.
Let’s assume all the remaining odds are 13hits only. The damage calculated this way will be lower than actual estimation. The idea is really to compare the damage output between line-up#4 and line-up#3: if the reduced damage of line-up#4 is still higher than line-up#3, then line-up#4 outperforms lineup#3.
(0.02821% * 16/16 + 1.449% * 15/16 + 98.523% * 13/16) * (146/73)^1.35 = 2.0824 times compared to line-up #1
If total damage per hit for line-up#1 is 100, line-up#4 would be no less than 208.24
Update:
I wrote a simple program to find the numerical solution for the odds (pure math is hard) and here is what I’ve got:
- if the hero is killed on turn #1, Kailani / Gunnar chameleon and Muggy chameleon may take up to 2 hits away from the 16 rounds. 97.3861% of the time titan will land 14 hits on your shielded heroes, 2.5836% for 15 and 0.0303% for 16.
- if the hero is not killed on turn #1, Kailani / Gunnar and Muggy chameleon may take up to 3 hits away from the 16 rounds. 90.4968% of the time titan will land 13 hits on your shielded heroes, 8.0287% for 14 , 1.4442% for 15 and 0.0303% for 16.
This gives us an accurate damage estimate for line-up#4 to be
(0.0303% * 16/16 + 1.4442% * 15/16 + 8.0287% * 14/16 + 90.4968% * 13/16) * (146/73)^1.35 = 2.0887 times compared to line-up #1
If total damage per hit for line-up#1 is 100, line-up#4 would be 208.87
I’m currently using this lineup and here is the damage for perfect situation: Gunnar was one shot killed on round 1, and all 16 round of titan attacks are reflected. Gunnar chameleon and Muggy chameleon were both killed by titan’s AOE skill. Damage towards a 14-star blue titan is 128936.
Conclusion
Line-up#4 has the best statical efficiency of using titan shields. Note the assumptions used for drawing this conclusion.
In reality my recommendation goes to line-up#3. Reasons being:
- statistical estimated damage output difference is negligible: between 2%~3%
- it’s very likely that you will accidentally stun the titan when shield is active (unwanted combos), when that happens you’ll lose damage from reflecting 1 or 2 hits or maybe even more. line-up#4 is mainly depending 13 reflections and line-up#3 is mainly depending on 15 hits. Losing 1~2 hits means more to line-up#3 than line-up#4: 1/13 compared to 1/15
- it’s also likely that you won’t be able to perform 16 rounds of actions. skill animation, unwanted combos all take precious time away from your titan fight. missing 1 or 2 turns will reduce damage output more for line-up#4 than line-up#3: 15/16 -13/14 is much less impact than 13/16 - 11/14.
