Ha... I know the feeling. I drafted a mage without any board clears or even a freaking frost nova. Lost really hard against a shaman using double flametongues and the +hp on totems card. Also lost against a pally playing a lot of small cheap stuff.
In that situation, where you don't have any board clears, going first or pulling a combo to get a minion advantage seemed pretty necessary to win. Being able to be the aggressor makes all the difference when you can't take away his minions easily.
Before I got in beta earlier this week, I watched a lot of YouTube vids of matches. Totalbiscuit isn't a terrible Hearthstone player, and he explains a lot of his reasoning. It's not always the best, but I felt like it at least gave me an edge once I got in. Trump is probably the best streamer on Twitch since he explains a lot of his reasoning and is great at the game.
Still, even though I've been watching for awhile, the best I've done in arena is 4-3. That shit's on a whole different level of play/metagaming.
Also, I'm kinda tempted to buy a BlizzCon digital pass for their tournament. It's constructed decks, but the players bring 3 decks and if they win they continue using that deck (so your opponent knows what's in it), and if you lose, you switch decks. Sounds like a lot of fun.
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