I have a deck of cards out of which 2 players randomly choose 6 each, and puts them on the table.
The cards are double sided and have 3 optional sides. A card can’t have the same the same option on both sides. So, if we have sides A, B and C, the cards are AB, AC and BC.
What should be the MINIMAL deck size to provide a good statistic spread, meaning each player will get every side at least 2 times, and avoid cases that all cards are facing up with the same side. There is an option of re-shuffle so even 80-90% chance of good spread is also acceptable.
Consider a deck of size $3N$ with $N$ each of $AB,AC,BC$. There are, of course, $\binom {3N}6$ ways to choose $6$ cards from the deck.
How many of these avoid an $A$ completely? Well, we need to get $BC$ six times so there is exactly $1$ way to do it. Similarly for faces $B,C$ so the probability of missing some face completely is $\frac {\binom N6}{\binom {3N}6}$. We remark that it is not possible to miss two faces completely.
Note: in the large $N$ limit the probability of missing a face completely is $$3\times \left(\frac 13\right)^6\approx 0.004115226$$
What's the probability that we get only one $A$? Well, we need $BC$ five times and then the remaining card can be either $AB$ or $AC$. As there are $6$ possibilities for the non-$BC$ card the probability of getting exactly one $A$ is $$2\times N \times \frac {\binom N5}{\binom {3N}6}$$ Of course the probabilities of getting exactly one $B$ or $C$ are the same so we must multiply this by $3$. Again we remark that it is not possible to get exactly one of more than one face.
Note: In the large $N$ limit the probability of getting some face exactly one time is $$3\times 6\times \left( \frac 13\right)^5\times \frac 23\approx 0.049382716$$
The probability of getting all face up $A's$ is $$\frac {\binom {2N}6}{\binom {3N}6}\times \frac 1{2^6}$$
This is fairly small, in the limit the probability of getting all face up $A's$ is $\frac 1{3^6}\approx 0.001371742$ so the probability that we get all face up sides matching is three times this or about $0.004115226$
Thus, in the large $N$ limit the probability that a given player gets a "good" hand is $1-.057613169=0.942386831$
Now, to search for a minimal $N$ that passes some threshold you need to specify that threshold and use the explicit formulas I wrote out.