By Neal - 8/18/2020
Strategies for 1-card draw, or Turn 1 Soitaire still generally apply, except that the cards in the hand are not all available; as you know, you can only play the third card you count out each time. So to the extent that you can, you need to control what cards become available and when.
If you go through the hand without moving any cards to either the columns or Ace piles, then the next time through, all the same cards, and only the same cards, will be available, because you’re just going to count out the same sets of three cards.
For example, in 1-card draw, we deliberately refrain from playing certain cards in the hand. Maybe we’re holding back the 5♣ in our hand, in hopes of uncovering the 5♠ in a column. Either black 5 could be put on a red 6 in one of the columns, but moving that 5♠ allows access to the card(s) under it in its column. But in 3-card draw, if you go ahead and take the 5♣ in your hand, and put it on that red 6, then, after that pass through the hand, you'll be counting out different sets of three cards, and maybe that’s more important than waiting for the 5♠ to show up in one of the columns. There's more detail below.
We have to reconsider our strategy in 1-Card Draw of never moving a 2 from the hand to the bottom of a column. If you move a 2 from the hand to the bottom of a column, there are no lower cards to move onto it from another column (so putting that 2 down doesn’t help expose any cards, whether you’re playing 1- or 3-card draw). Also, the 2 covers at least the 3 that it's on, until the Ace shows up for that 2. So normally, it's not a good idea to move a 2 from the hand to a column.
But when you’re playing 3-card draw, taking a 2 from the hand may be the only way to change the sequence of cards that will be drawn afterwards, as discussed above (and in painstaking detail below).
When you deal out three cards, it's not a good idea to play all three of those at once. If you play just one or two, then in the next pass through the hand, you will have different cards available for play. But if you play all three that are dealt out at one time, then you haven’t changed which cards get counted out afterwards. It's better to go ahead and play just one or two, rather than all three, so that in the following passes through the hand, you have different cards available to you.
Not only is it a bad idea to play all three cards that get dealt out at one time, but it can also be a bad idea to play a total of three cards in one pass through your hand - or six cards, or any other multiple of three.
Consider the 24 cards left after the seven columns are laid out. They become the hand, to be dealt out in eight instances of three cards. We'll number each card: card number 1 through card number 24. ALSO, those numbers remain with each card in the hand until the end of the game. So at some point you may have twelve cards in your hand, but they still have the original numbers that were assigned to them, anywhere from 1 through 24.
Before the first pass through the hand, the card numbers that we know may actually be played will be cards 3, 6, 9, 12, 15, 18, 21, and 24 - and more, but only if any of these are played.
Let's walk through this, and for an example, look specifically at the third and fourth instances of dealing out three cards. That's six cards, and they're identified, distinct from all the other cards in the hand, and for the duration of the game. These two instances of three cards are card numbers 7 through 12. Before them are cards 1 through 6, and after them are cards 13 through 24.
Okay, with that background, suppose that in your first pass through the hand, you move only card numbers 9, 8, and 12 to the columns or Ace piles.
Then through the rest of that first pass, you don't take any more cards from the hand.
So, in the second pass through the hand, in the first two instances of dealing out three cards, you'll still deal out the same cards, numbers 1 through 6 (and the playable cards are card numbers 3 and 6).
Then the next three cards will be the original card numbers 7, 10, and 11, and you're free to play number 11, if there's a place to put it. (Of course, this is because cards 9, 8, and 12 were played earlier.)
NOW - here's the key to this entire exercise: afterwards, you will see the same original, and playable, card numbers 15, 18, 21, and 24. And you may find that you still can't play any of them, which means of course that you still can't get to any of the cards below them.
Here's the fix: if, in that first pass, you had played only the original card numbers 9 and 12, or perhaps only card number 9 or 12, then during the second pass, nearly every card you see will be different. Instead of number 15 you'll see number 14; instead of 18 you'll see 17; and so on.
Now this is likely what they call counting the cards, but it is what you need to do in this game: avoid playing three cards in a single pass, or six, or any other multiple of three.