What are viable strategies for trying to win Shadows over Camelot? Our group always seems to lose when we play. :(
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Shadows Over Camelot is actually unbalanced towards the Knights winning. In fact, the base game has even been solved (guaranteed victory) when playing with 7 players. The key to that victory lies in the special white cards. Since they're reshuffled (both decks are reshuffled when you reshuffle one) when you run out of cards, what the Knights should do is keep their hands mostly full as often as possible, thus they can quickly cycle Merlin cards and other whites, instead of 'saving' them.
Therefore, when starting the game, everyone should spend time drawing up white cards at Camelot, until they reach their limit of 12. Only then should someone go questing. The only exception to this is Lancelot's armor. The first knight who has a full house should immediately go claim the armor, as it's the most powerful tool for white victory in the game.
Because of this strategy, the most important Knight roles are King Arthur (for passing cards) and Sir Galahad (for playing the special whites for free). In general, Arthur gives the specials to Galahad, then he stays at Camelot drawing them and playing them (Merlins included).
Remember, this is a race against time, so movement is the enemy. Every time you move to a quest, you should lose a life to perform the quest's heroic action.
Now that bad blacks, life (through piety), cards, and even redrawing (through convocation) are dealt with, all that remains is how to complete quests.
As I said, the Armor is the most important. Get that quickly. Ignore the Dragon, for the most part. When fighting Picts and Saxons, your goal should be to complete the quest with 2 or 3 picts/saxons on the board, since then, effectively, those would have been 'wasted' black cards (i.e. don't be too safe and not put any mercenaries on the one you're working on, and, of course, don't lose the quest). Try to keep the Grail and the Sword, JUST SHORT of victory, until the armor is attained. If you complete these quests too early, you'll just be adding catepults. While this is happening, always take black cards, as they're the easiest. Cats are much more expensive to deal with.
Once you've gotten the armor, and defeated a pict/saxon or two, then gang up to get the sword, draw back up on grails, and then gang up to finish the grail. Since black cannot 'gang-up' on a quest, you're sure to have victory with the overwhelming numbers. You do the Grail last (hopefully with a Heroism card on it), and get 4 white swords for it. (Heroism is key to winning as well). The traitor won't matter, since even flipping two won't affect the outcome. Don't waste time on accusations. The first person to accuse will be the traitor, so if he accuses, accuse him back and it's dealt with.
Late in the game (during the gang-up phase). Use life points and cats to avoid drawing the black cards.
In terms of special blacks, most aren't that bad. The ones to really worry about (i.e. use 3 merlins to stop) are Mists of Avalon (always), Vivian (always), and Dark Forest (if the grail is in danger of being lost, only).
Here's how a typical game might go:
Traitor unrevealed (8 white 6 black)