After 3 straight days of sarging I can open easily enough now. Last week approach anxiety was a huge issue; now it's almost non-existent. But I still have a lot of trouble carrying the conversation on.

A typical approach (based on the last few days):

*RedBaron approaches the set*
RedBaron: Hey, I need your opinion on something. I've got to get to my friends in a second, but blah blah some kind of opinion opener blah
Her: blah
RedBaron: blah blah blah
Her: lol blah
RedBaron: So what are you doing here/shopping for/buying/etc?
Her: I'm looking for a vibrator/ I'm here checking out dudes to see who has the biggest bulge, and you're the winner/ I'm looking for edible underwear to bring home to my hot lesbian lover/ Etc.
RedBaron: blah blah
Her: blah blah

At this point, either the set will begin to walk away, or they'll stay where they are but I'll run out of things to say and eject out of the set.

So I guess what I want to know is, how do you guys advise moving the conversation forward? The above example, with a little extra fluff talk, was enough to get me a number close, but I want to take my game farther than that.

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D.G.A.F. isn't just a state. It's a lifestyle.

I have never done day game. But I think is a bad idea to use indirect game during day. Problem is, you have very little to work with as far as building attraction goes. Also, people have goal to get to point a to point b so your limited with time.

Textbook says bitch shields are usually down during day since these girls don't get approach that much, so it's better to go direct.

If you use opinion opener, you might be coming across as some weirdo or insecure. At least with direct, you come across as being confident instead of beating around the bushes.

Also, if you notice a pattern, a lot pua who teach day game teach direct style. You could open with something like "hey I thought you were cute, just wanted to say hi." or "I gotta get your opinion real quick, ok, I don't I just wanted to talk to you because I thought you were cute."

I know this does not answer your question. But you might wanna try to go direct, then if your still running out of things to say, come back and ask the same question again.

Im not gonna lie half of the time a girls talking i'm interacting but still thinking of things to say or asking her to elaborate on some different things she talked about

I agree with this. Girls will even be flattered and interested by a guy who goes direct during the day, so they'll make more of an effort to continue the conversation to see what kind of guy you are.

You got her number right? If so give her a call the next day. You already showed you have balls by getting her number. Take her out and talk to her then. You dont want to spend to much time with her right when you get her number she might be busy and in a hurry. Give her a call then start with the fluffing convos.