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1v1 Constructed
Only the most recent sets
Standard
Everything since Return to Ravnica
Pioneer
Everything since Eighth Edition
Modern
All cards, but with many bans
Legacy
All cards, very few bans, some cards are restricted to 1 copy
Vintage
Only commons
Pauper
Play with a "leader" against 1 opponent (It's just the other format name + 1v1)
Commander 1v1
The only rule is your deck must be at least 40 cards and your sideboard no more than 15 cards. Other than that, you can play whatever you want and have as many copies as you want.
Freeform
Limited
choose a card then pass the pack and choose from the next
Draft
Open 6 packs and build from that
Sealed
choose and pass, but with a "leader" (two other format names put together)
Commander Draft
Things happen while you choose cards
Conspiracy
Multiplayer
Choose a "leader" for your deck
Commander
A special card changes the game rules
Planechase
Your "leader" is a planeswalker instead of a creature
Oathbreaker
Arena/MTGO
Eventually this will become the same as a paper format
Explorer
All cards on arena with some changes and bans
Historic
Most recent sets with some changes and some digital only cards
Alchemy
Play with a "leader" but only cards from recent sets
Brawl
Play with a "leader" but only cards on Arena
Historic Brawl
All cards on Arena. No bans, but some cards restricted. Only digital-only cards have been changed.
Timeless
All your cards are basic lands. Summon random creatures.
Momir Basic
Teams
Normal constructed 2v2
Two Headed Giant
Choose cards alone, then play with teammates (team + the 1v1 format name)
Team Draft
Open 12 packs together and build 3 decks (team + the 1v1 format name)