A game for any plural number of players:
Every player "gets" the numbers, twice each, of 1 to n -- where n is some pre-determined positive integer -- to play with; the integers 1 to n placed in each player's "mod pile", and the numbers 1 to n placed in each player's "add pile".
(A pile may simply be a list of integers that the player crosses off as each integer is played. Or if n = 13, for instance, and there are two players, then there can be 4-total actual piles where the numbers are represented by playing cards of certain suits.)
When the game begins, we have the initial value of the variable m, m(0), equal to 0.
The players take turns moving.
On a move, the player moving can choose a number from either their mod pile or their add pile. Let this number be c(k), when on the kth move of the game. Each number can only be picked once from a given pile. If playing with playing cards, place the card representing c(k) in the "used pile". Only pick from cards in your mod or add piles, not in the used pile.
On the kth move of the game, the player, depending on whether they chose their number from the add pile or the mod pile, sets m(k) to equal:
m(k) = m(k-1) + c(k), if c is chosen from the add pile.
m(k) = m(k-1) (mod c(k)), if c is chose from the mod pile.
0 <= m(k) <= c(k)-1.
If, and only if, c(k) was chosen from the mod pile, then m(k) is added to the moving player's score.
Play until the numbers from both piles of all the players are picked.
(There will be 2*n*(number of players) moves.)
Largest score wins.