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I’ll tell you what my guess was!
I entered “griffin”/”griffon/”gryphon””~! After all, “big cat” = lion and griffins have a bird’s head and wings too~! :3 (Er, I submitted it as “griffin” but I just wanted to point out the different spellings. ^^)
I sure hope the animal is cute cuz that dragon isn’t and the cat sure isn’t either! I think I’ll do my list of cute Tiny Farm animals one day soon. XD
Edit on March 14th: I’m noticing people coming to this page while searching for info on how to get the Radiant Egg.
It all begins with a Tiny Egg! The process is like this: Tiny Egg hatches a Silver Tiny Egg; Silver Egg hatches a Golden Tiny Egg; a Golden Tiny Egg hatches a Radiant Egg. Basically, you’ll be spending Bells buying Tiny Eggs to gamble that they’ll eventually turn into a Radiant Egg. The problem is that, somewhere along the way, an animal may hatch instead! 😡
So while it’s fine and dandy that we have 2x the chance to get a Pegasus until March 20th , what Com2uS SHOULD have given players is 2x the chance to get a bloody Radiant Egg.
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When will Com2uS gives us a break? Well, I suppose the frequency of quests are all right since it’s not like there’s much else to do in the game. (And they’re completely optional too so I guess I shouldn’t complain. :B)
First part of the quest is to give the Phoenix some love as the description reads, “Give Love to Phoenix in Alfredo’s or your friend’s farm once”.
I hope that Tiny Egg reward will give me a Winter Alpaca because I’m still missing it! 😡 But I also wouldn’t mind a super cute baby Phoenix. It’s so adorable. *^_^*~ Sadly, it won’t stay as a chick forever. 😦
Is it just or do the designs of the Phoenix and dragon look like they belong in another game?
I suppose I’ll update this post as I progress through the quests. I wonder if I’ll manage to get just one of the Legendary Animals? I really want the Phoenix just so I can have the chick. XD
Update~!
Second quest is to give the legendary Fox some love and third quest is to love the Dragon. Both of them can either be at Alfredo’s farm or at a neighbour’s. Not really related to the quests but it looks like the Silver Tiny Egg is available for purchase in the Shop. I suppose Com2uS mentioned this everywhere except in the notice screen when the game loads up.
Incidentally, it looks like the Jewelry Tiny Egg has been renamed to Radiant Egg. I kind of liked it more when Tiny Farm had the 3 types of eggs. >:/ Now we have to buy the Tiny Egg (or Silver Tiny Egg) and hope that it’ll hatch to become a Radiant Egg. 9__9 Guess I should have gotten more Frozen Eggs when I had the chance to try for the Winter Alpaca. 
Oh yeah, from the two quests that I already completed, I got a Grey Polar Bear and a Talking Spotted Pig from the eggs. 😐
Update #2:
Fourth (and last?) quest is to find a Black Pegasus and give love to that. Of course, Alfredo doesn’t have it … It’ll probably show up after the promo though. As usual, the Com2uS Tiny Farm forum has a list of users that have the animals you need, including the Black Pegasus. However, keep in mind that there are thousands of people trying to groom them too. My suggestion is to add as many of these users as you can and you may luck out eventually.
The Tiny Farm hub is also a good source during these events~! Good luck, everyone!
koukoupuffs 