This puzzle is sort of a duck konundrum (a puzzle where you follow a series of instructions to get the final result, leading to a solution). However, the end result has already been given, we just need to find out what to do with it. There are two steps to finding the solution here.
Each card suit has a unique way of being presented, with a custom backing color and face. The custom faces can be connected to decks from The Deck of Many Things. The color of the backs are picked based on the name of the deck, just for theming. Each suit has its own deck associated with it.
The colors represent ports and batteries. The lines give which row to use, and the number gives which column to use (following the arrow).
Each zig-zag of hexagons can be seen as a "column". The number on the top gives which column to use, and the highlight points out a specific cell which gives the card.
Multiply the number of dots by the number of hands. This gives a value to use in the table, giving our card.
The pip given is the actual pip of the card. The suit given tells us which circle to use. The orientation of the pip gives a direction in the circle, providing the actual suit of the card.
As the rule card explains, each suit has a rule assigned to it which alters the value of a card. There are hints on the rule card which can help us determine the suits rules, but we can also use the story to find these values. For example, we know that the first pair Mark placed has a value of 11.
|Clubs||(Evens only) +4|
|Diamonds||+[# of cards is discard]|
|Spades||(Odds only) Reverse order|
*Because the value for the diamonds would change, the value shown is what was used in the game. Cells that have "N/A" were unused cards.
Clicking the value of each card shows what the card would look like.
Once the values of each card have been found, we take the values of each player's hand at the end of the game and convert them to letters. Mark's final hand appears to be backwards, but if we look at it from his side of the table it's the right way around.
I have also made a github page which contains all the card designs and backings for the community to use linked here. Be sure to look at the README file. :)