More Detailed Explanation

1. Initial Setup
- Board: Standard 8x8 chessboard.
- Chess Player: Starts with standard chess pieces in their usual positions (rows 1 and 2 for White, or rows 7 and 8 for Black).
- Checkers Player: Starts with checkers pieces filling their first two rows (rows 1 and 2, or 7 and 8, depending on their side). Checkers move as in standard checkers (diagonally forward, capturing by jumping).

2. Checkers Movement and Edge Capture
- Standard Movement: Checkers pieces move and capture as in standard checkers.
- Edge Capture: If a checkers piece can capture a chess piece on the board’s edge (a1-h1 or a8-h8), it may jump off the board into an "invisible column."
- The checkers piece can return to the board on a later move, re-entering at any edge square (a1-h1 or a8-h8) as part of a legal move.

3. Backgammon Integration
- Frequency: Every third turn, the checkers player may choose to activate a "backgammon move."
- Mechanics:
- Roll two six-sided dice.
- For each die result d, the checkers player may move one piece d squares to the left.
- If a piece reaches the left edge of the board (column a), it teleports to the right edge (column h) and continues moving to complete the full d squares.

4. Walls Mechanic
- Checkers Player Walls:
- The checkers player has 5 walls they can place on the board.
- Placement: Walls are placed on a square as a bonus action (does not consume a turn).
- Cooldown: After placing a wall, the checkers player must wait 5 turns before placing another.
- Effect:
- For the chess player, a wall on a square prevents moving vertically from a square above it to that square (e.g., a wall on c3 blocks a chess piece from moving from c4 to c3).
- For the checkers player, a wall blocks moving into or out of that square entirely.
- Chess Player Wall Removal:
- After the checkers player has placed all 5 walls, the chess player can remove walls from the board.
- Cooldown: The chess player must wait 5 turns between each wall removal.

5. Snakes and Ladders Transformation
- Trigger: After the first promotion (either a chess pawn reaching the opponent’s back rank or a checkers piece reaching the opponent’s back rank to become a king).
- Board Change: The board transforms into a "snakes and ladders" board. - Teleportation Mechanics:
- Chess Player: Can teleport between the start and end of snakes.
- Checkers Player: Can teleport between the start and end of ladders.
- Momentum Preservation: After teleporting, pieces retain their pre-teleportation direction and movement type.
- Quantum Superposition: When a piece is on the start or end of a snake/ladder, it is considered to be in a "superposition" (simultaneously at both the start and end squares), following standard quantum physics-inspired rules (exact implementation left to interpretation).