16 Players Preditcion Problem

Hello all! For an activity I'm organizing, I have about 16 players. Each player have interaction with 6 others - 3 which on his plus side, 3 which on his minus side (he can't have a partner that's on both sides).
Basically, what I'd love to know is if the number matches and in the end each player would have 6 partners. I have no real knowledge on math, so I'm sorry if it's a dumb or obvious question. I tried arranging that with tables, thinking that it would probably need 18 players, but I always seem to need more players, to a never ending amount.

Just to clarify, I'm talking about something like that:
Let's look at the 7 first players: Matt, Andrew, Beth, Robb, Zoey, Sean and Daisy. Another note, the 1st player could have interaction with all of the others, potentially, including even the 16th player. I just write it as a follow up for your convenient.

Matt
Plus side: Andrew, Beth, Robb.
Minus Side: Zoey, Sean, Daisy.

Andrew
Plus side: Matt (predetermined because it was decided on Matt's list), Beth, Robb.
Minus side: Zoey, Sean, Daisy.

Beth:
Plus: Matt, Andrew (predetermined), Robb.
Minus: Zoey, Sean, Daisy.

Now we've reached a point where Zoey, Sean and Daisy (Numbers 5, 6 and 7) has no more people on their minus side, so we have to "jump" with Robb to numbers 8, 9 and 10 to continue the count.

Robb:
Plus: Matt, Andrew, Beth (predetermined).
Minus: N8, N9, N10.

When we reach Zoey (N5), her plus side is open, so we can still use N8, N9 and N10 (either on the plus or minus side, both are alright as it's still open).

Zoey:
Plus: N8, N9, N10 OR N11, N12, N13.
Minus: Matt, Andrew, Beth (predetermined).

And so on...

Click to expand...

So, I hoped I made myself clear, as English is not my native language. Will every player have 6 others (3 plus, 3 minus) or do I need to add/subtract players?

Thank you so much ahead,
Thijs