Final Fantasy IV - Farming for Rainbow Pudding, Tails, and Hidden Summons

 The final subquests of Final Fantasy IV consist entirely of things to grind for.  The items you receive will be retained in a New Game Plus, so the effort won't go to waste.  I'd suggest doing this once you have the Treasure Hunter augment, which you receive by completing every in-game map.  This means you'll have completed everything in the game except defeating the final boss, you'll be able to travel everywhere, and you'll have lots of gil built up.

The items you need to farm for are as follows;

• Rainbow Pudding: to complete the -ingway Quest.
• Hidden Summons: to complete Rydia's summon list, which earns an achievement.  The hidden summons are Cockatrice, Bomb, Goblin, and Mindflayer.
• Tails: to trade for Onion equipment and Adamant Armor. 

All of these will be retained across a new game plus.  The -ingway Quest will remain exactly where you left it.  Unused summon items, untraded tails, and the Onion equipment and Adamant Armor will remain in your inventory.  Rydia's summon abilities will remain, once learned.  You might also choose to farm for Silver Apples, Gold Apples, and Soma Drops.  While these items are retained in a New Game Plus, the effects of using them aren't.  Save them for your last playthrough.  Green and Yellow Dragons drop Silver Apples as well as their respective Tails, so you'll be picking some up while you're farming for those.

Generally, to farm for an item, you'll go to a specific location and use a Siren to summon a battle.  Sirens can be purchased from the only town on the moon, in the southeast cave.  In some locations, the Siren will always summon one specific battle.  In other locations, the Siren may summon up to four different battles.  Lunar Cave East is the one place you will not want to use a siren, as the battle summoned does not contain the enemies you're hunting.  Instead, you'll need to fight random battles here.

This table shows each item you'll need to farm for, why, and where you can find it.

Item	Purpose	Equipped By	Dropped By	Location
Pink Tail	Trade for Adamant Armour	Final Five	Flan Princesses	Use a siren in the most southwest cave of the Lunar Subterrane B5.
Rainbow Pudding	-ingway Quest	Quest
Red Tail	Trade for Onion Sword	Cecil and Kain	Red Dragon	Use a siren on B8-B10 of the Lunar Subterrane.
Black Tail	Trade for Onion Armour	Final Five	Armoured Fiends	Use a siren on B2 & B3 of Lair of the Father
Green Tail	Trade for Onion Helm	Final Five	Green Dragon	Use a siren just outside the entrance to the Sealed Cave
Yellow Tail	Trade for Onion Gloves	Final Five	Yellow Dragon	Use a siren just outside the entrance to the Passage of Eidolons
Blue Tail	Trade for Onion Shields	Cecil and Kain	Thunder Dragon	Use a siren on B2 or B3 of the Passage of the Eidolons
Cockatrice	Hidden Summon	Rydia	Helldiver, Cockatrice, or Fledgling Rukhs	Use a siren outside Baron. 25% chance you'll face only Helldivers
Goblin	Hidden Summon	Rydia	Goblins, Lil' Murderers, or Mopingway	Use a siren inside the Mist Cave
Mindflayer	Hidden Summon	Rydia	Mindflayer	Use a siren inside the passage between B3 and B4 in Lodestone Cavern
Bomb	Hidden Summon	Rydia	Balloons or Dark Grenades	Battle in Lunar Cave East 25% chance you'll face all bomb-types 25% chance you'll face 2 bomb-types plus another enemy

Note: When I first wrote this, this table was an image taken from my spreadsheet like all the others.  However, because of the size, it was difficult to see the rightmost column.  I later started experimenting with better table designs.  I improved this one but not the others because the others were fully readable from the start.

When you complete a battle, the game randomly picks which enemy will drop an item, and then chooses an item (or 'nothing') from that enemies drop list.  Generally, in this game, an enemy has four potential items to drop, with odds as shown below.  Having the Treasure Hunter augment active doubles the odds of any item dropping.

The table below shows the odds of winning each item if you battle as described above.  

This table is the chance you will have won the item after battling X amount of times.  E.g., if 1000 of us sat down to try to win the Mindflayer item, around 100 of us would have it within 14 battles, and 800 of us would have within 206 battles.

I found these calculations useful because it let me set goals like "I'll do 200 battles (or until I get the item) then do something else".  I would find a goal like "I will fight endless battles until I win the item" impossibly frustrating.  As I type, I'm still hunting for Pink Tails, Red Tails, Black Tails, and Blue Tails, and my goal for today is to reach a total of 206 battles for each enemy.  If there are still any left to win after that, my next goal will be 294 battles.  Then I'll probably go up in increments of 50, though I really hope it doesn't come to that.  These are all just tricks for my sanity, the odds are still 0.8% for every battle.  

Finally, this table is how many battles I actually fought to win each item.  It will be updated when I've finished farming.  These are battles I specifically counted while farming and does not include the times I just happened to fight those enemies while playing through the game.  For items like Bomb, when sometimes the battle wouldn't even be against the right enemy, I still counted every battle, not just those against bomb types.

EDIT:  My final count.