Senet OpenAI Gym

• Rules
  • Kendall
  • Skyruk
• Installation
• Environment
  • Observation
  • Actions
  • Rewards

---

Rules

Senet is a board game for two players. It is played on a board which consists 3 rows of 10 squares. Squares are known as "houses".

Some houses are marked with hieroglyphic pictograms and have special meaning:

• 15: marked with ankh 𓋹, is the Rebirth House
• 26: marked with 𓄤𓄤𓄤, is the House of Happiness, or Pretty House
• 27: marked with water sign 𓈗, is the House of Water
• 28: marked with three ba birds 𓅢, is the House of Three Truths
• 29: marked with two women 𓁐 with crowns, is the House of Re-Atum, or House of Isis and Nephthys
• 30: marked with flacon 𓅃, is the House of Re-Horakhty

Each player starts with 5 or 7 pieces, which are known as "dancers". Due to specific shape of the pieces, they are referenced as "cons" for one player, and as "spools" for another. The goal of the game is to guide own dancers along a specific path and finally bourne them all off the board before the opponent does so. Dancers can be bourne off the board once they reach one of five special houses: House of Happiness, House of Water, House of Three Truths, House of Re-Atum or House of Re-Horakhty.

Dancers are moved from their staring houses to the last house along the board following an S-shape path. Namely, from left to right in the upper row, from right to left in the middle row, and, finally, from left to right again in the bottom row. At the beginning dancers are placed in alternate houses starting from first house along the line of their movement. That is, for the game with 5 pieces, "cons" are placed in houses 1, 3, 5, 7 and 9, whereas "spools" are placed in houses 2, 4, 6, 8 and 10. First move takes the "spools" player.

In each turn one dancer can move forward or backward to a given number of steps. The game is equipped with 4 sticks, one side of each is black, another side is colored. The player having a turn throws these sticks. The number of steps is the number of sticks come up colored sides. If all sticks come up black side, the number of steps is 5. Player obtains an extra move if the number of steps is 1, 4 or 5. Therefore, the turn is passes only if the number of steps is 2 or 3.

Normal dancer move direction is forward. However, if no dancer can move forward, player should move one of the dancers in backward direction to the same number of steps. If no dancer can move neither forward no backward, the turn is passed to another player independently of what number of steps was. No more move backward is allowed for dancers which reached House of Happiness

No house can be occupied with more than one dancer. Dancer can not move to the house where another dancer of the same player is located. If, however, the landing house is occupied with dancer of another player, and the move is legal, the dancers swaps their houses: the opponent's dancer is moved to the house of depart of the player's dancer. In this case Player's dancer beats the opponent one.

Such move is legal if landing house is not safe and the opponent dancer is not protected. The safe houses are all special houses: House of Rebirth, House of Happiness, House of Water, House of Three Truths, House of Re-Atum, and House of Re-Horakhty. If two dancer of the same player stand next to each other, they protected each other. Depending on particular rules, some other combinations of dancers are considered as protected. Also in some cases it is not legal for dancer to jump over some special groups of opponent's dancer, the blockades. The exact description of blockade depends on particular rules.

There are lot Senet rules variations, we consider two of them:

• simple and most known ones, well adopted for algorithmic implementation, we reference it Kendall rules, although the original rules proposed by Tim Kendall are more complicated.
• less known rules by Dmitry Skyruk, which we reference Skyruk rules, which probably have deeper connection with original Egyptian game.

Below we explain what blockade and protected dancer mean in each version of Senet rules, and how to move when landed to a special house.

Kendall

• Blockade: three or more opponent's dancers in a row, all dancers should stand next to each other
• Protected: two or more opponent's dancers in a row, none of dancers within protected group can be beaten
• Special Houses:
  • House of Happiness is mandatory. Dancer is not allowed to move further if this house is not visited. If, for example, dancer is in house 25, next to the House of Happiness, the player should while number of steps is exactly 1 to move the dancer to the House of Happiness
  • For a dancer in the House of Happiness:
    • if number of steps is 5, the dancer bourne off the board
    • if number of steps is 4, 3, or 2, it is moved to the House of Re-Horakhty, House of Re-Atum or House of Three Truths respectively
    • if number of steps is 1 it is moved to the House of Water, but...
  • The House of Water is a trap, the dancer which lands here gets drawn in the water, and reappears in the House of Rebirth. If House of Rebirth is occupied, the dancer is moved to the first empty house preceding the House of Rebirth
  • The dancer in the House of Three Truths can be bourne off the board if the number of steps is 3, this is the only legal move for this dancer
  • The dancer in the House of Re-Amun can be bourne off the board if the number of steps is 2, this is the only legal move for this dancer
  • The dancer in the House of Re-Horakhty can be bourne off the board if the number of steps is 1, this is the only legal move for this dancer

Skyruk

• Blockade: two or more opponent's dancers in a row, all dancers should stand next to each other
• Protected: when two or more dancers stand next to each other, the first and the last one are protected. Unlike Kendall rules, the inner dancers are not protected and can be beaten
• Special Houses:
  • House of Happiness is mandatory. Dancer is not allowed to move further if this house is not visited. If, for example, dancer is in house 25, next to the House of Happiness, it lands to the House of Happiness whatever the number of steps is.
  • Once a dancer reaches the House of Happiness, sticks should be thrown again and this dancer performs an extra move according to the thrown number of steps
    • if number of steps is 1, the dancer is moved to the House of Water, if House of Water is occupied, dancer is moved to the House of Rebirth, if House of Rebirth is occupied, the dancer is moved to the first empty house preceding the House of Rebirth
    • if number of steps is 2, 3 or 4, the dancer is moved to the House of Three Truths, House of Re-Amun or House of Re-Horakhty, if the landing house is occupied, the dancer is moved to the House of Water instead, if the House of Water is occupied as well, the dancer is moved to the House of Rebirth or to the first empty house preceding the House of Rebirth
    • if number of steps is 5, the dancer is bourne off the board
  • Unlike Kendall rules, the House of Water is necessary is a trap, instead
    • if number of steps is 1, 2 or 3, the dancer is moved to the House of Rebirth or to the first empty house preceding the House of Rebirth
    • if number of steps is 4, the dancer is bourne off the board
    • if number of steps is 5, the dancer remains in House of Water and the turn is passed to the opponent
  • The dancer in the House of Three Truths can be bourne off the board if the number of steps is 3, this is the only legal move for this dancer
  • The dancer in the House of Re-Amun can be bourne off the board if the number of steps is 2, this is the only legal move for this dancer
  • The dancer in the House of Re-Horakhty can be bourne off the board regardless the number of steps.

---

Installation

git clone https://github.com/mecotrade/gym-senet.git
cd gym-senet/
pip install -e .

---

Environment

Observation

Observations come as a pair of the board itself and the player having next move. The board is encoded and a numpy array of shape [2, 30], where 0-th component is for the cons, and 1-th component is for spools. The values are 1 if the house is occupied with a dancer of a given player, and 0 otherwise. The empty house has therefore [0, house] == 0 and [1, house] == 1.

Actions

There are 3 equivalent representation for actions:

1. Action is represented as a pair (house, num_steps) where house is the house index, from 0 to 29, whereas num_steps is number of steps, positive for move forward and negative for move backward. When no move is possible, neither forward not backward, the dumb action is represented by pair (30, 0).
2. Another way to represent an action is to use dancer index instead of house index, more specifically, all dancers of given player is enumerated from starting house to the house of Re-Horakhty. Important point, the index is not referenced to a specific dancer, instead, to its relative position. The action is represented, therefore, by a pair (dancer_index, num_steps). When no move is legal, this dumb action is represented by (-1, 0).
3. Finally, any pair of type (dancer_index, num_steps) can be converted to a single number from the interval [0..10 * num_pieces], that is, [0..50] for 5 pieces game and [0..70] for 7 pieces game, according to the following formula:
   • dancer_index * 10 + num_steps if num_steps = 1, 2, 3, 4, 5
   • dancer_index * 10 + 11 - num_steps if num_steps = -5, -4, -3, -2, -1
   • and 0 stands for the dumb action

The third way is suitable for the MCTS implementation.

Rewards

Rewards is +1 when "cons" player wins, -1 when "spools" player wins, and 0 after any move in the middle of the game.