As you know, Shooting the Moon in Hearts is when you manage to take all the Hearts and the Queen of Spades... if you don't know that, you really should read the rules first.
There's a lot of strategy behind successfully Shooting the Moon, but most of it is in correct planning. If you plan the game correctly and pass the right cards, then most of the game should play itself.
We should first cover what kind of cards you actually need to attempt this. There are a few possible combos. I'll try to put them in to a few groups and write what other special things you need in each case and how to play.
This situation is really easy to play. Before the hand you should pass the low cards that you have, in hopes of getting even more high cards. The only thing that can totally destroy your game is, if you get a bunch of low hearts during mulligan. In that case, you should just abort the mission unless you also have a bunch of the highest Hearts (you should have at least J - A). In case you don't have the highest Hearts, you should keep their number to 2 max (pass the rest, if you have to).
The play itself is easy. Lead out with any low cards of suits that have not been played yet to get rid of them. In case you don't have the highest Hearts (for example, you have the King and the Jack of Hearts), you should lead them as soon as possible. This way you'll give yourself the best chance to take them all down, since other players won't know what you are doing and will still be avoiding Heart tricks.
This situation is a little trickier. It's absolutely a must to have the highest Hearts. The more of all the Hearts you have, the less of the highest ones you'll need. For example, if you have 10 Hearts, it's enough to have just the Ace and King (you'd have to get really unlucky to not pick them all up in this situation). Another important thing to have is the Highest cards of all or most other suits. This will ensure that you pick up the trick in which the Hearts will get broken. It's also really cool, if you have the Ace or King of Spades.
Playing can be a bit tough. You'll need to make sure that you pick up the trick in which the Hearts are broken. This is easiest to do, if you have 4 or more high cards of one suit. Then you just lead it out 4 times and there's a good chance someone will break Hearts. Once they are broken it gets really easy. All you have to do is lead out the high Hearts and take them all down.
There are two neat tricks you can use during play. You can lead out with the Ace or King of Spades as soon as you can. At that point other players won't know what you're up to and will gladly give you the Queen.
The other trick is to play your highest Hearts from the lowest to highest. You should do this because, if you lead out with the Ace right away, it becomes pretty obvious what you are doing and other players will start saving their highest Hearts in hopes of spoiling your plan.
This is probably the easiest situation to play, but you need to play smart. Make sure you have the highest cards of that suit and at least 7 cards. Also make sure you don't have many cards of another suit, unless you have mostly high cards. You can also have a few Hearts, bu they have to be between J - A and if they are not all the highest, make sure you don't have more than two.
So basic play strategy is to lead out with low cards of your non dominating suits (to get rid of them while everyone still has that suit) and keeping the high ones so that you can take control of the game later. If you have Hearts, but not the highest ones, then lead them out as soon as you can (while others still don't know what you're doing and won't spoil your plans). So once you are left with only your dominating suit and all the highest cards, just start playing the highest cards of that suit and you should pick up every single trick up until the end of the game.
Be very careful not to put yourself in a situation where you can't take control of the game! This can happen pretty easily since most of your cards consist of one suit.
That's it... Now go and Shoot the Moon!
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