Enigma Sphinx

If a spell with cascade is countered, the cascade ability will still resolve normally.

The mana value of a split card is determined by the combined mana cost of its two halves. If cascade allows you to cast a split card, you may cast either half but not both halves.

When the cascade ability resolves, you must exile cards. The only optional part of the ability is whether or not you cast the last card exiled.

If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.

Cascade triggers when you cast the spell, meaning that it resolves before that spell. If you end up casting the exiled card, it will go on the stack above the spell with cascade.

If you have zero or one card left in your library when Enigma Sphinx's middle ability resolves, Enigma Sphinx is put on the bottom of your library.

After Enigma Sphinx's middle ability triggers, if Enigma Sphinx is removed from your graveyard in response, it won't be put into your library.

If you control an Enigma Sphinx that's owned by another player, it's put into that player's graveyard from the battlefield, so Enigma Sphinx's middle ability won't trigger.