"Clean" and "dirty" typically refers to whether or not a lion has any inbreeding in their heritage.

Inbreeding has no effects, but many people avoid it either to create an additional challenge or because it grosses them out, and some avoid it purely because the market for clean lions is better.

The "Big 4" are a set of kings with an incredibly high cub count, mainly due to their immense stats. I don't remember their names, though. Most (if not all?) of them are inbred anyway.

Whether or not a lion is "good" is almost entirely dependent on the player. If you care about how much they're worth to sell, a good lion would have a lot of breed-only features, or be a complete clone of a raffle lioness. But if you as a player love the Brawl and Rumble marking appearances, you can go ahead and fill your pride with them if it makes you happy!

Mutations have a low % of showing up randomly without using any items - they are usually common ones like Achromia or female mane mutations. The chance is higher when the fertility is lower. Rarer mutations like hybrids or Dwarfism can show up as well, but it's very very unlikely. Cotton Root Bark, Lion Meat, and GMO Cows will all increase the chances of any mutation showing up. Lion Scrotum gives a 25% chance of the mother's mutation passing, or 5% if the mutation is Double Uterus. It doesn't work on hybrids.

There are several ways to earn currency:
Participate in events and sell event items (the price rises later in the year when those items are more scarce)
Get lucky with breeding (popular BO base+marking cubs can sell for quite a bit, or mutations)
Hunt+sell food (slower but steady income)
Studding

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