Anna’s Day at the Arena

This is a puzzle about the Fire Emblem series of games. Each of the four arenas references a specific game in the series, sorted by chronological order of release:

• Original Arena refers to Fire Emblem: Shadow Dragon and the Blade of Light. (Original clues the original NES version of the game, not the remake for the DS.)
• Sacred Arena refers to Fire Emblem: Sacred Stones.
• Dawn Arena refers to Fire Emblem: Radiant Dawn.
• Triple Arena refers to Fire Emblem: Three Houses.

In each arena, a battle log is given, detailing the combat between the four characters. Using the stats and the battle mechanics of each game, we can reverse engineer the missing stat for each of the characters. We give potential solvepaths in the appendix.

Upon determining everyone’s missing stats, we notice that the characters are ordered by their missing stat from left to right, cluing that we should reorder. We order the characters in each game alphabetically.

Extracting via 1-26 to A-Z gives the cluephrase ANSWERWARSTORIES, leading to our final answer of WARSTORIES.

Appendix: Potential solvepaths

Original Arena

We can notice that the second combat occurs if Barst hits Julian and defeats Julian in one hit. We infer that character 3 is Barst, character 2 is Julian, and Julian has 9 HP after the first combat.

Next, we look at what scenarios would mean character 1 was one point of strength off of defeating character 4. This only makes sense from Caeda being character 1 and Jagen being character 4 (as one point would allow a crit to kill).        

From here on, we iterate through all possible choices of the missing stats and match with the percentages to get the final stats:

Battle Results:

1. Caeda hits, Julian misses. Julian loses 14 HP.
2. Barst hits, Julian is defeated.
3. Caeda crits, Jagen crits, Caeda is defeated. Jagen loses 27 HP.
4. Barst misses, Jagen hits, Jagen hits. Barst loses 22 HP.
5. Jagen hits, Barst is defeated.

Sacred Arena

Note: For this arena and all following arenas, probabilities reflect True Hit. In most Fire Emblem games after FE 6, the random number generator generates two numbers for every attack, and compares their average against the visible hit rate. This generally causes high hit/miss rates to hit/miss more often, and biases the numbers in favor of the player (as player characters tend to have higher accuracy). This is referenced in Anna's first note.

The two combats where the speed was involved (almost certainly - it's possible that accuracy is so low that AS/dodge becomes involved but by computing some preliminary hit rates that is clearly not possible) involves combat at the AS threshold (i.e. the AS difference between the combats must be 3 or 4). In particular, character 2 must have 3 more AS than character 3, and character 1 must have 4 more AS than character 4.

In particular, this tells us that Seth and Lute are not character 1/4 or character 2/3. Lute has 5 AS and Seth has 14 AS. Neimi's AS is equal to her speed, and Tana's AS is at minimum 7 and maximum 20.

We do casework on character 3.                                                

• If character 3 is Seth, then character 2 must be Neimi or Tana, and character 4 must also be so (since Def does not change the result of combat with Lute). Thus, in this case, character 1 here would be Lute, and so character 4 must be Neimi and character 2 must be Tana (due to AS). In particular, Seth would be hit by one hit by Tana first, then one hit by Neimi. There is no way for this to occur where 1 more Def prevents Seth from getting killed, so this case cannot occur.
• If character 3 is Tana, for AS reasons, character 2 cannot be Lute. Similarly, because Lute uses magic, character 4 cannot be Lute. Thus character 1 must be Lute, meaning that for AS reasons, character 2 must be Seth and character 4 must be Neimi. In particular, Seth would be hit by one hit by Tana first, then one hit by Neimi. There is no way for this occur where 1 more Def prevents Seth from getting killed, so this case cannot occur.
• If character 3 is Neimi, for the same reasons as before, character 4 cannot be Lute. Moreover, for AS reasons, character 1 cannot be Lute. Thus, character 2 must be Lute. Thus, Lute has 5 AS and Neimi has 2 AS, Seth has 14 AS, and Tana will have 10 or 18 AS. If character 1 is Tana, then in the first combat, Tana can attack Lute twice. The character's hit rates and crit rates are determined, and trying all possibilities, no possible combats match battle 1. If character 1 is Seth, then Seth has a 36 critical rate against any other character and has at minimum a 101 hit stat, while Lute has 10 avoid. Thus, Seth has at least a displayed 91 hit rate, or at least a 98.47% chance of hitting Lute. No such possible combat possibility would lead to an 86.5% combat probability, as the probability of a miss, crit, or non-crit hit by Seth on Lute are all below 86.5%. Thus, there are no possible choices for character 1, so this case is impossible.

We conclude that character 3 must be Lute. Then character 2 has 8 AS (and thus is not Seth). Moreover, character 2 will hit Lute once (and not crit, given the probability), and character 4 must attack Lute twice. character 2 must be defeated after initiating against character 4, which can only happen if character 2/4 are Seth/Tana in some order. Thus character 1 is Neimi.                                                

Thus, characters 1-4 are Neimi, Tana, Lute, and Seth, respectively. From here on, we iterate through all possible choices of the missing stats and match with the percentages to get the final stats.

We also note that if extraction is noticed early (by ANSW in the first arena, for example), a lot of this work can easily be bypassed by guessing Lute’s Def as 5 and Neimi’s Spd as 18.

Battle results:

1. Neimi hits, Neimi hits. Tana loses 28 HP.
2. Tana hits, Lute hits. Tana loses 7 HP, Lute loses 15 HP.
3. Seth hits, Lute hits, Seth hits. Lute is defeated. Seth loses 19 HP.
4. Tana hits, Seth crits. Tana is defeated. Seth loses 26 HP.
5. Neimi hits, Neimi hits. Seth is defeated.

Dawn Arena

The only way the second combat could occur must involve a double Astra (in order to have so many events leading to such a small probability), so one of the two combatants must be Mia. Mia cannot proc Astra on Leanne (Nihil) and does not have the speed to attack Naesala twice, so the two characters involved are Mia and Oliver.

The only character with a 100% hit rate is Leanne (and cancels all skills due to Nihil, so no need to worry about those activation rates), so character 4 must be Leanne. Between Mia and Oliver, Leanne has a 100% hit rate only on Oliver, so character 2 is Oliver, character 3 is Mia, and character 1 is Naesala.

From here on, we iterate through all possible choices of the missing stats and match with the percentages to get the final stats:

Battle results:

1. Vantage activates. Oliver hit, Naesala hit, Naesala hit. Naesala loses 26 HP, Oliver loses 48 HP.
2. Vantage activates. Oliver hit. Mia activates Astra. Mia hits five times, Oliver activates Pavise five times. Mia activates Astra. Mia hits five times, Oliver activates Pavise five times. Mia loses 30 HP, Oliver gains 30 HP.
3. Leanne hits. No damage taken.
4. Mia crits, Naesala hits. Mia is defeated. Naesala loses 21 HP.
5. Naesala activates Tear. Oliver is defeated.
6. Leanne hits. Naesala is defeated.

Triple Arena

In the fourth combat, the only two characters who would lose AS from 1 strength is Catherine and possibly Lysithea, if her strength is 0 mod 5. If character 3 is Catherine, for her to lose one strength and now be attacked twice, her opponent has to be Claude (and thus Claude’s AS - Catherine’s AS = 3 before the strength loss). However, this can only happen if Claude initiates combat, and in that case, Claude would be character 3. Thus, this cannot happen, so character 3 must be Lysithea, and character 2 must be Seteth.

If Catherine is character 4, there is no way for combat to play out such that one point of resistance would cause her to survive. Thus, character 4 must be Claude, and character 1 must be Catherine.

From here on, we iterate through all possible choices of the missing stats and match with the percentages to get the final stats:

Battle results:

1. Catherine hits. Catherine hits. Seteth hits. Catherine hits. Catherine hits. Catherine loses 29 HP. Seteth loses 28 HP.
2. Lysithea activates Crest of Gloucester and hits. Claude misses. Claude misses. Claude loses 30 HP.
3. Claude hits. Claude hits (Desperation). Seteth hits. Claude is defeated. Seteth loses 26 HP.
4. Vantage activates. Seteth crits, Lysithea activates Miracle. Lysithea hits (no crest). Seteth is defeated. Lysithea loses 40 HP.
5. Catherine misses. Catherine misses. Lysithea misses (no crest). Catherine crits. Lysithea is defeated.

Authors' Notes

Why does Fire Emblem have so many mechanics?

This puzzle was a ride. I originally had a mathy answer in mind, so my thought was “What if I combined Fire Emblem and math into a puzzle”, and then we got this monstrosity. In the end, we didn’t even use the mathy answer (although the answer we got still was very thematic).

Earlier versions of this puzzle had testsolvers do data collection on all of the weapons, abilities, and skills. Unfortunately, the first wiki that usually shows up when you google Fire Emblem stuff is the Fandom.com wiki, which is notoriously terrible for Fire Emblem. Here is a list of things that the Fandom.com wiki got wrong while this puzzle was being testsolved:

• The names of the weapons Lance and Bow for FE1 were incorrect.
• Paladins were not listed as a horseback class for FE8.
  • I made this change on the wiki, and they reverted it. For no reason. Cool.
• The Iron Bow page made no mention that they are effective against flying units.
  • I know this is common knowledge for FE players, but for non-FE players, it might be easy to miss this if you just look on the Iron Bow page.
• The Faire skills are listed incorrectly for FE3H (+5 Attack is correct, not +5 Might).
• The description for Defiant Crit was incorrect (it is now fixed).

It also doesn’t help that other, more trustworthy wikis don’t always list all of the properties of the Daemon Card, which leads to some issues.

The line about an “idealized random number generator” was needed because (from what we could tell) FE1 uses an 8-bit seed, which significantly affects probabilities. Other games appear to use a 32-bit seed for their RNGs, and this potentially causes issues with the really small probability event we have. We decided to just avoid this issue entirely.

Thanks to Justin Yokota for helping check a lot of the facts for this puzzle.