Increasing Ambition — ruling 1

"Flashback [cost]" means "You may cast this card from your graveyard by paying [cost] rather than paying its mana cost" and "If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack."

Increasing Ambition — ruling 2

You must still follow any timing restrictions and permissions, including those based on the card's type. For instance, you can cast a sorcery using flashback only when you could normally cast a sorcery.

Increasing Ambition — ruling 3

To determine the total cost of a spell, start with the mana cost or alternative cost (such as a flashback cost) you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined only by its mana cost, no matter what the total cost to cast the spell was.

Increasing Ambition — ruling 4

A spell cast using flashback will always be exiled afterward, whether it resolves, is countered, or leaves the stack in some other way.

Increasing Ambition — ruling 5

You can cast a spell using flashback even if it was somehow put into your graveyard without having been cast.

Increasing Ambition — ruling 6

If a card with flashback is put into your graveyard during your turn, you can cast it if it's legal to do so before any other player can take any actions.

Increasing Ambition : règles, fonctionnement et interactions (MTG)

Increasing Ambition is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

Search your library for a card and put that card into your hand. If this spell was cast from a graveyard, instead search your library for two cards and put those cards into your hand. Then shuffle. Flashback (You may cast this card from your graveyard for its flashback cost. Then exile it.)

How it works : Increasing Ambition

Original English text — Wizards of the Coast / Scryfall

#1March 19, 2021· Wizards of the Coast
"Flashback [cost]" means "You may cast this card from your graveyard by paying [cost] rather than paying its mana cost" and "If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack."
No other card is named in this ruling.
#2March 19, 2021· Wizards of the Coast
You must still follow any timing restrictions and permissions, including those based on the card's type. For instance, you can cast a sorcery using flashback only when you could normally cast a sorcery.
No other card is named in this ruling.
#3March 19, 2021· Wizards of the Coast
To determine the total cost of a spell, start with the mana cost or alternative cost (such as a flashback cost) you're paying, add any cost increases, then apply any cost reductions. The mana value of the spell is determined only by its mana cost, no matter what the total cost to cast the spell was.
No other card is named in this ruling.
#4March 19, 2021· Wizards of the Coast
A spell cast using flashback will always be exiled afterward, whether it resolves, is countered, or leaves the stack in some other way.
No other card is named in this ruling.
#5March 19, 2021· Wizards of the Coast
You can cast a spell using flashback even if it was somehow put into your graveyard without having been cast.
No other card is named in this ruling.
#6March 19, 2021· Wizards of the Coast
If a card with flashback is put into your graveyard during your turn, you can cast it if it's legal to do so before any other player can take any actions.
No other card is named in this ruling.