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if you wait for the events that have alot of different rarity eggs in the reward pool on the wheel it will pay off way better, with the sunday mass event your better off saving your diamonds and gambling to make them grow.

You can acquire the flaapy eggs from the event spins for diamonds if you cant do the flaapy game.

Common flaapy eggs: Charmander, Cleffa, Diglett, Electrike, Geodude, Hoothoot, Ledyba, Machop, Magikarp, Mareep, Marill, Nidoran (both), Oddish, Phanpy, Pidgey, Poochyena, Rattata, Sentret, Slugma, Spinarak, Sunkern, Swinub, Treecko, Zigzagoon, Zubat.

I also acquired a shiny Electrike from a common flaapy egg.

Cracked around 150 rare flaapy eggs, so new additions are.  Telepup, Magneton, Nidorina, Strikewolf, Pixiekit, Magmar, Crobat, Stantler, Larvitar, Fearow, Aggron, Donphan, Vulpix, Riolu, Vibrava, Medicham, Tentacruel, Ursaring, Rhydon, Dunsparce, Rockruff, Qwilfish, Clefable, Mr. Mime, Delibird, Shadepup, Shadowkit, Pikachu, Xatu, Dodrio, Quilava, Azumarill, Wartortle, Chinchou, Mienshao, Eevee, Tauros, Exeggutor, Lickitung, Sparkpup, Venompaw, Wigglytuff, Politoed, Weavile, Torchic, Octillery, Absol, Rapidash, Machamp, Luxio, Axew, Mudkip, Froslass, Ambipom, Happiny, Seadra, Skarmory, Weezing, Poliwhirl, Zorua, Sandwolf, Poliwrath, Mamoswine.

Epic:  Steelpaw, Flareon, Vaporeon, Jolteon

As far as I know, no one has gotten a male of any of the fnaf event mons.  I will also point out that the event only mons can NOT be gotten from eggs, breeding the fnaf mons usually gets you metangs or jirachi.

Epic Flaapy eggs: Magmortar, Leafeon, Glaceon, Steelix, Kabutops, Lapras, Kangaskhan, Aerodactyl, Yanmega, Ditto, Pupitar, Heracross, Raichu, Lucario, Gardevoir, Wraithing, Umbreon, Luxray, Voidwolf, Porygon2, Zoroark, Dragonair, Alakazam, Gengar, Thunderlord, Espeon, Shelgon, Togetic, Scizor, Electivire, Rhyperior, Tangrowth

Rare flaapy eggs: Corsola, Slakoth, Kingler, Marowak, Jumplff, Mindkit, Ivysaur, Cloyster, Hitmonlee, Forretress, Scyther, Tidalfin, Nidorino, Munchlax, Persian, Electrode, Granbull, Lairon, Glalie, Mismagius, Ampharos, Kadabra, Dratini

Carmelita - (Hydreigon + Tyranitar) (Poochyena + Umbreon)

Michiru - (Renamon-Nun + Riolu)

Ankha- (Latias + Latias) (Trapinch + Terrabeast) (Celebi + Terrabeast)

Blaze - (Umbreon + Houndoom) ( Magcargo + Flareon)

Fifi - (Spinarak + Gengar) (Spinarak + Knight-Nun)

Gumy - (Sylveon-Love + Cleffa) (Vaporeon + Marill)

Chansey the preevolve of blissey is from an epic, scalepup, the first preevolve of the line that goes Scalepup, Drakefang, Titanwyrm is also from an epic flaapy egg.

Munchlax is from Rare and is the preevolve of Snorlax which comes from common flaapy eggs.

Your right, my mistake the hitmonchan was from a rare egg.  I've gotten remoraid, igglybuff, shellder, mankey and doduo from common eggs.  From what I can tell the eeveelutions are good ways to get carmelita, michiru, blaze, and gumy.  Ive gotten blaze from Umbreon +Houndour and Magcargo + Flareon.   Gumy from Leafeon + Venusaur and Leafeon + Vileplume.

I'm trying to make sure not to list any flaapy egg results that can be gotten through breeding.

common flaapy eggs:  Psyduck, Grimer, Koffing, Meowth, Caterpie, Metapod, Weedle, Kakuna, Sandshrew, Seel, Venonat, Goldeen, Azurill, Spearow, Bellsprout, Voltorb, Krabby, Hitmonchan, Horsea, Omanyte, Paras, Tentacool

Your not the only one that can't access those images, this is a known issue and has been reported on their bug list so assume they are working to fix it.

https://itch.io/t/6442498/unofficial-guide-tips-and-research 

If you want the tldr version then try either 2 Celebi together or Renamon-Nun + Cleffa.  Both of those breeds have a chance for 2 different multiverse mons.

Braixen-Nun has the Fairy type and that is how you got Gumy

I bred a batch using Cleffa + Renamon-NUN and got 2 Gumy and 2 Michiru. Later got another Gumy from Cleffa + Sylveon and I can verify that Tawna Bandicoot is only available from Epic level Flaapy eggs.  2 Celebi bred together can give you both Ankha and Gumy.  Weezing + Ekans gave me Fifi le Fume.

Carmelita Fox from Hydreigon + Gible

Michiru from Latias +Gallade

Gumy from Vileplume + Sceptile