When I was younger, I was a student at the National Mathematics Summer School (NMSS). We spent most of our time doing two things: mathematics and games. Sometimes it was hard to tell the difference. Certainly, there was one game we played, called Mod 13, where the two mixed in energetic bursts of mental arithmetic and chaotic shouting.

The game is a fun way to learn modular arithmetic, one of the core topics taught at the school. In fact, I believe the game was invented at the school, inspired by the lectures. Here is how to play it.

The rules

The game uses a standard 52-card deck. Start by dealing 7 cards to each player. Place the remaining cards face down to form the draw pile.

Flip over the top 2 cards from the draw pile to start the discard pile. From here on, any player with a legal card can discard it from their hand onto the discard pile. Players can play at any time in any order.

A legal card is one for which the number of the card is congruent modulo 13 to the result of one of the allowed arithmetical operations. For this purpose, aces are treated as 1, Jacks as 11, Queens as 12 and Kings as 13 (which, of course, is congurent to 0).

The allowed mathematical operations all operate on the previous 1, 2, or 3 cards in the discard pile. They are as follows:

• Unary operations
  • Multiplicative inverse (‘inverse’)
• Binary operations
  • Sum (‘sum’)
  • Product (‘product’)
  • Arithmetic progression (‘AP’)*
  • Geometric progression (‘GP’)
• Ternary operations
  • Sum (‘sum of last 3’)
  • Product (‘product of last 3’)
  • Quadratic progression (‘QP’)**

When you play a card, you must shout out the name of the operation you are using. The standard phrases are shown in the brackets above. For example, if the top two cards are 3 and 7, a possible legal set of moves is as follows (starting with the first two cards):

 3
 7
10 (sum)
 4 (sum)
 1 (product)
 2 (sum of last 3)
 7 (inverse)
 2 (inverse)
 2 (product of last 3)
 4 (product)
 6 (AP)
 8 (AP)
 1 (sum)
 5 (GP)
12 (GP)
 4 (sum)
 7 (QP)
10 (sum of last 3)

*For AP, there’s a limit on how many times it is allowed to be used consecutively. It is as many times as the absolute value of the common difference in the AP. For example, if the top card is 6 and the one below it is 5, then you can place a 7 and shout ‘AP’, but no one can then continue it (because the common difference is 1). Another example is if the top card is 5 and the card below it is 2, then you can place an 8 as an ‘AP’, and it can be continued twice more (the next one is 11 and then 1), but then no more. If the common difference is -1, then you can still only continue the sequence by one card because the absolute value of the common difference is 1.

**A QP can only be played if the sequence being generated is a quadratic progression but not an arithmetic progression (it needs to be a ‘proper’ QP). This is to prevent players from circumventing the AP restriction described above.

If a player makes an error, and someone points it out, then the player must take their card back and also draw an extra card from the draw pile as a penalty.

If there is ever a long pause in the game (~10 sec for experienced players, longer for beginners), it is standard assume that no one has a legal card to play. At that point, everyone draws one penalty card, the current discard pile is set aside, and a new discard pile is started by flipping over the next 2 cards from the draw pile.

The first player to discard all of their cards wins the current round.

A single round of Mod 13 is usually quite short. A full game consists of multiple rounds. The winner of each round starts with one extra card in subsequent rounds, which accumulates with each win. The first player to win a round after starting with 13 cards wins the game.

Tips and variations

• If you run out of cards in the draw pile, pause the game to replenish it from the discard pile. Resume the game by drawing the top 2 cards, like at the start of a new round.

• You can play this game with any number of players. Simply add more decks of cards until you have enough. You can also have any number of draw piles, simply divide and spread them around so everyone has one within easy reach.

• You can vary the length of the game by changing the starting number of cards. For example, starting with 5 cards rather than 7 makes the game longer, since more rounds are required for someone to get to 13 cards. You can also change the target number to be smaller or greater than 13 cards.

• You’ll find that it’s easier to do the mental arithmetic if you think of a Queen as -1, a Jack as -2, a 10 as -3, etc.

• The game becomes quite chaotic as you increase the number of players and as the players increase in skill. This happened without fail at NMSS. Each round became more and more rapid, making the shuffling and dealing between rounds quite tedious in comparison. To maximise gameplay, we got lazy and invented the NMSS shuffle. This involves flipping all the cards face down and everyone helping to spread them around vigourously. Then everyone simultaenously picks up cards at random to form their hands, and helps to gather the remaining cards to form one or more draw piles.