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As you already know, flipped cards will flip all cards its arrows point to onto its new color, and you flip cards if you have an arrow pointing to them.
Onto strategy.
The AI doesn't really know how to handle high defensive cards. Alexandria, Ribbon, and especially Meteor are really good for this. Alexandria is a good first move, as the AI will often try to attack it and fail. Meteor is incredible against some opponents (Weapon Master Hunt lacks any magic cards, and so can't hit its weakness) and really bad against others (Eidolon Master Leyra with all the magic). However, it has very high attack and physical defense.
Try not to leave exactly one avenue to create a card battle on a given card. If there are two unused arrows, let the AI make the first attack/blocking move. This will allow you to flip the card without possible retaliation, and in turn take the AI card 'for free'. This is another place where Alexandria shines- if you have an A card, then it will attack the weak attack stat and be an easy retake that the AI quite literally can't use.
http://steamcommunity.com/sharedfiles/filedetails/?id=665845408
Your card has arrows. If you place your card down next to an opposing card that has no arrow facing your card, you automatically flip it to your color. If the card does have an arrow facing your card, your card attacks it and the two engage in a battle. Whoever wins the battle (or ties, if you're the defending card) flips the opposing card to its color. If you win and your opponent's card happens to have arrows pointing at other cards, those cards will also be turned to your color (no battle necessary) as part of a combo.
Who wins a battle depends on stats and lots of dumb dumb double ♥♥♥♥♥♥♥ dumb RNG.
First symbol = attack strength (from hexadecimal 0 to F)
Second symbol = attack type (P, M, X, or A)
Third symbol = physical defense (same as attack)
Fourth symbol = magical defense (same as attack)
Attack and defense numbers aren't exact. If you know your hexadecimal, then think of the attack stats as the first digit in a two-digit hexadecimal number. The second digit is randomly generated whenever a card does battle with another card. So a card with 0 attack has anywhere between 0 (00) and 15 (0F) actual attack strength. (If you don't know your hexadecimal, then consult the chart in the above guide.)
The attack types are as follows:
P = physical attack, will always target an opposing card's physical defense value
M = magical attack, will always target an opposing card's magical defense value
X = "Power" attack (above guide calls it flexible), will target the weaker of the two defense values
A = "Advanced" attack (above guide calls it assault), will attack with the highest value on the attacking card against the lowest value on the defending card irrespective of whether those values are attack or defense values (1A54 will attack with its 5 rather than its 1, for instance)
The generated attack value of the attacking card and the generated defense value of the defending card are the numbers you see when the cards do battle. The game then generates ANOTHER set of random numbers, one for each card and neither of which are visible to the player, that is anywhere between (or equal to) zero and that first generated number. It subtracts the second number from the first, and whichever card has a higher remainder wins. In the case of a tie, the defender wins.
Let's say Card A attacks Card B.
Card A = 0P00
Card B = 0P10
Card A actual attack value (from RNG) = 10 (0A)
Card B actual defense value (from RNG) = 20 (14)
Card A second RNG = 4
Card B second RNG = 18
Card A first value minus second value = 6
Card B first value minus second value = 2
Card A wins.
So even though Card B has the higher defense value, because the RNG rolls a disadvantageous second number, it loses. If the second number had been anywhere from 0 to 14, Card B would have won.
One last note: Cards can upgrade their stats to a certain extent. The mechanics of it are as dumb as the card game itself, but suffice it to say that a 0P00 card has a chance of upgrading to 2X43 or so if you keep winning with it.
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It's this reliance on RNG that makes the card game the abysmal mess that it is. Even the number and placement of arrows on the cards are randomly generated. I'd love a mod that fixes this minigame. Make the arrow placements fixed for each card, and simplify the stats. Keep the "leveling up" feature if liked, but eliminate the RNG calculations so that a 4 always beats a 0.
This (freakishly powerful) mod tool, named "Hades' Workshop," might be of interest. I haven't seen anyone use it to touch the card game, but I'm sure it's possible. People have used it to mod in Beatrix as a fully recruitable character; a card mod would be easy peasy in comparison: http://forums.qhimm.com/index.php?topic=14315.0
Imagine if you were playing poker and you had to check the RNG if your flush beat the opponent's two pair.
Gripes aside.
Can I fully ignore this game? Or am I going to be forced to go through it at some point later?
You can avoid it with no negative consequences whatsoever EXCEPT for a mandatory one-time story card game. If you win, you get a relic/add-on (which I believe becomes available soon thereafter). If you lose, you don't get it. That's it.
Then you have to play a third game right after, where the prize for winning is a Rebirth Ring. Now, you can synthesize Rebirth Rings later, but one of the synthesis materials for it is finite within the game world, and it's expensive to boot. You defintely want to win this.
Other than that one point in the game, which lasts about 5-10 minutes, you can completely ignore the cards.
You don't need to farm cards for this point in the game, as every enemy has a 12.5% chance to drop a card on defeat and there are many available treasures. Further, if you have less than 5 cards, you can talk to Doctor Tot and he will give you enough cards to reach 5- and these are generally pretty strong cards, and you can repeat this process until you have a good set of arrows.
Hurray.