# Numerica Rules

Numerica is a mathematical board game between two players, where the aim is to place tiles on a 15x15 board in order to construct mathematically valid equalities. An example of a valid equality is:

6 = 4 × 3 − 6.

An example of an invalid equality is:

2 = 3 + 1.

1. When you invite to a game, you can choose between a standard board or a random board. For a standard board, all the special cells with point multipliers (2P, 3P, 2E and 3E) are in the same place every time. For a random board, the special cells are placed at random each and every time.

2. When an invitation to play Numerica has been accepted, the player to start off the game is selected at random. The first tiles to be placed on the board must cover the middle cell of the board.

3. The aim of the game is to construct mathematically valid equalities, either horizontally (reading left to right) or vertically (reading downwards). For each move, you must form at least one complete set of valid mathematical equalities (e.g. 2 = 1 + 1 = 3 − 1), and no tile can be placed in such a way that it forms a mathematically invalid expression in combination with its neighbours. For instance, you cannot place a tile marked '×' next to a tile marked '='.

4. Some of the adjacent tiles to the ones you place may form incomplete equalities. An incomplete equality is an equality where one of the members is left empty. This means you are allowed to form adjacent equalities such as 6 + 2 =, but not 6 + 2 = 9 −. Note that you do not gain points for incomplete equalities.

5. Expressions are evaluated according to the usual operator precedence rules, that is, multiplication and division is done before addition and subtraction, so that 1 + 6 × 2 equals 13, not 14.

6. Existing equalities on the board can be extended. So for instance, you can place tiles to extend 2 = 1 + 1 to become 22 = 1 + 19 + 2. However, repetitions are not accepted. So you cannot extend, for instance, 3 + 3 = 6, to 6 = 3 + 3 = 6. Also, "leading zeros" and "leading plus signs" are not accepted. This means that expressions such as 06 = 3 + 3 and +3 = 2 + 1 are rejected. When you extend an existing equality, you score for the entire extended equality, but point multipliers are not re-applied if they've already been applied in a previous move.

7. At least one of the tiles you play must be placed adjacent to a tile which is already placed on the board. The exception is the very first tiles placed on the board, which are instead required to pass through the middle cell. You can modify existing equalities on the board as long as you do not create mathematically invalid expressions.

8. Each player will receive 9 tiles to start, and you will refill up to 9 tiles when you use tiles in the game, until there are no more tiles left to take from. There is a total number of 110 tiles in each game, distributed as follows:

#### Points

0 7 1
1 7 1
2 7 2
3 7 3
4 7 3
5 7 4
6 7 4
7 7 5
8 7 5
9 7 8
+ 7 2
4 3
× 7 4
/ 4 5
= 16 1
Blank 2 0

9. Each tile gives a certain number of points (see table above), and the point sum of an equality will be the sum of the tile points. When all tiles are used the game is over. You can also resign when it is your turn. This will end the game and you will lose, regardless of the distribution of points.

10. You have 72 hours to play an equality. If you exceed this limit, you will lose the game.

11. A blank tile is a joker, and can represent any given number or mathematical symbol, but it does not give any points.

12. 2P stands for Double Points. Any tile put on a 2P cell will contribute the double of the value of the tile to the total score for the equality.

13. 3P stands for Triple Points, and will triple the points of that specific tile.

14. 2E stands for Double Equality count. If you manage to cover a 2E cell with your equality, your total score for that equality will be double of the value of all tiles used.

15. 3E stands for Triple Equality count. This will triple the score of your equality.

16. Multipliers on individual tiles are applied before considering word multipliers.

17. If you manage to use all 9 tiles available in 1 play you trigger a "BINGO". This adds 50 points to your total score for the equality.

18. If you manage to create an equality of fractions, you will score additional points according to the following rule: The fraction will first be reduced to canonical form (i.e., irreducible with positive denominator) - for instance, 4 / − 6 becomes − 2/3. The additional points gained are 20 multiplied by the denominator minus 1. For instance, − 2/3 gives an additional 40 points (2 × (3 − 1)).

19. If a player is unable to form new valid mathematical expressions with the tiles at his or her disposal, he or she can instead choose to pass his or her turn to the opponent. If three passes are made in a row, the game will end and the player with the most points wins.

20. A player can also choose to swap one or more of his/her tiles instead of playing an equality. It is not possible to swap more tiles than the number of tiles available in the bag.

21. The game ends when one of the players has used all his or her tiles and there are no more tiles left in the bag, known as playing out, or after three consecutive passes. If a player plays out, the sum of the values of his/her opponent's tiles is added to the player's score, while the opponent's score is reduced by the same amount.

## A Note on Rating Scores

Numerica uses an Elo inspired system for rating players: Each player gets an initial rating of 1500. After completing a game, the new rating of a player becomes:

Rnew = Rold + K(Sreal − Sexp)

where:

 Rnew = new rating Rold = old rating K = is a constant equal to 32 for rating < 2100, equal to 24 for ratings between 2100 and 2400, and equal to 16 for ratings above 2400 Sreal = is the actual score, i.e., 1 for a victory, 0.5 for a tie, and 0 for a loss Sexp = is the expected score, equal to the probability of the player winning the game plus half the probability for a tie ( = 0.333 + 0.167 = 0.5 if both players have the same rating)

The formula used to compute the expected score is, given players A and B:

Sexp(A) = 1 / (1 + 10(Rold(B)-Rold(A))/400)