Contraconquer.
Version of Thursday 17 January 2019.
Dave Barber's other pages.

Introduced here is a playing-card game of the same genre as the classic Rock-Paper-Scissors (RPS). As with RPS, there is no element of chance, and all players have an equal opportunity to win.

Although Contraconquer can be played in different ways, this report starts with a three-player version using nine different cards, which might be printed thus:

A minimal game can be played with 9-card pack with 1 instance of each card, but a more interesting game results from using a multiple such as a 36-card pack with 4 instances of each card.

The large numeral on its card is its rank. The small numerals, included merely for convenience, indicate which combinations this card will win. As this version of the game is limited to single-digit numbers, the hyphens might be omitted as below:

To begin the game the cards are not shuffled. Rather, player A takes all the cards marked 'A', the other players similarly.

With a small loss of convenince, ordinary playing cards can be substituted. Player A takes the aces, twos, and threes; B takes the fours, fives, and sixes; C takes the seven, eights, and nines. A total of 36 cards will be in use.

The game is played as a succession of tricks. Each trick is a contest of cards, one from each player. Players contribute cards to tricks simutaneously:

• Each player selects a card from their hand and lays it face down on the table. All players do this at the same time, or approximately so.
• After all have done that, each player turns their card face up, again at more or less the same time.
• Players decide who won the trick, as explained below.
• Finally, the three cards of the trick are removed from play.

The chart below indicates who wins each possible trick. Equivalent information is printed on the cards themselves.

 combination winner 1-4-7 2-4-8 3-4-9 1-4-8 1-5-9 1-6-7 1-5-7 1-6-8 1-4-9 1-5-8 2-5-9 3-5-7 2-4-9 2-5-7 2-6-8 2-4-7 2-5-8 2-6-9 1-6-9 2-6-7 3-6-8 3-4-7 3-5-8 3-6-9 3-6-7 3-4-8 3-5-9 1 2 3 4 5 6 7 8 9 player A player B player C

Some players may prefer to consult the following table, which contains the same information:

 1-4-7 → 1 A 2-4-7 → 7 C 3-4-7 → 4 B 1-4-8 → 4 B 2-4-8 → 2 A 3-4-8 → 8 C 1-4-9 → 9 C 2-4-9 → 4 B 3-4-9 → 3 A 1-5-7 → 7 C 2-5-7 → 5 B 3-5-7 → 3 A 1-5-8 → 1 A 2-5-8 → 8 C 3-5-8 → 5 B 1-5-9 → 5 B 2-5-9 → 2 A 3-5-9 → 9 C 1-6-7 → 6 B 2-6-7 → 2 A 3-6-7 → 7 C 1-6-8 → 8 C 2-6-8 → 6 B 3-6-8 → 3 A 1-6-9 → 1 A 2-6-9 → 9 C 3-6-9 → 6 B

The game ends when players have run out of cards. A player's score is the number of tricks that they have won. Two or three players might tie in the number of tricks won, but no individual trick can be tied.

The game is designed so that all cards are equally valuable in capturing tricks. Further, the pattern of winning cards is as scattered as possible; here is a characteristic example. If within a trick, player A's card is a 1 and player B's is a 4:

• if C's is a 7, then A wins (1-4-7):
• if C's is a 8, then B wins (1-4-8);
• if C's is a 9, then C wins (1-4-9).
To state it generally, whenever only two cards of a trick are known, it is not possible to determine which of the three players will win the trick.

All players are equally likely to win the first trick; after that the odds will vary from trick to trick.

When played with a pack of the minimal 9 cards, each player can play their 3 cards in 6 different permutations. With 3 players, this amounts to 216 = 6 × 6 × 6 different games. Each player will take all tricks in 18 of them; in the other 162, each player will take 1 trick for a 3-way tie. Because of the high probability of a tie, most players will choose to use a pack of more than 9 cards.

For other than three players, the scheme detailed above is not always readily adaptable. For one thing, there might be too many winning combinations to fit on the printed card. However, there is a generic approach to suit any number of players, using cards of very plain design that contain only the rank. Here are examples:

For instance, four players would use ranks 0 through 3; seven players 0 through 6. In every case, the quantity of different ranks equals the quantity of players.

In a minimal game each player would start with one card of each rank, but multiples will probably be preferred. All players start each game with equal hands.

As above, play is a succession of simultaneous tricks. To find the winner, a bit of arithmetic is required:

• the ranks of cards are added;
• the sum is divided by the quantity of players;
• the quotient is ignored, but the remainder indicates the winner of the trick:
• remainder 0 means that player A wins the trick;
• 1 → B;
• 2 → C;
• 3 → D;
• and so forth.

Here are some example tricks from a six-player game, using ranks 0 through 5:

cardssumremainderwinner
0-1-3-3-4-5164E
1-2-3-4-4-4180A
0-0-0-1-1-1 33D
2-2-2-3-3-5175F
0-1-1-2-2-2 82C

In deciding the winner of a trick, any group of cards whose sum equals the number of players, or a multiple of it, can for convenience be discarded. In a six-player game, for instance, from the trick 0-1-3-3-4-5 can be removed 0, 3-3, and 1-5. This leaves 4, which is the final value. From 1-2-3-4-4-4 can be removed 1-2-3 and 4-4-4 (sum 12), leaving nothing, which means 0.

Standard playing cards can be used when necessary. In order to give all players an equal opportunity to win, not all of the standard ranks can be used. Because calculations are based on the remainder from division, the number printed on the card can often be replaced by a smaller number to simplify calculation, as listed in the table below; but this substitution is not required for correct play.

actual
rank
would
be
players
2345678
A11 111 111
220 222 222
331 033 333
440 104 444
551 210 555
660 021 066
771 132 107
880 203 20
991 014 3
10100 120 4
J111 235
Q120 000