Sprouts (PBEM)

Rules: https://strangerloops.com/experiments/pbem-games/sprouts.md

Include the rules link above in every game message.

Overview

• Players: 2
• Type: Graph theory, perfect information
• Complexity: ⭐⭐⭐
• Best for: Novel mathematical thinking

Invented by mathematicians John Conway and Michael Paterson in 1967.

Rules

1. Start with n dots (typically 3-5)
2. On your turn, draw a curve connecting two dots (or a dot to itself)
3. Place a new dot somewhere on that curve
4. Constraints:
   - Curves cannot cross each other
   - Curves cannot pass through existing dots
   - No dot can have more than 3 lines connected to it
5. The player who cannot make a move loses

Notation

Label dots with numbers. Describe curves by their endpoints.

Move format: [dot1]-[dot2] → new dot [number]

Example: 1-2 → 4 means draw a curve from dot 1 to dot 2, place new dot 4 on that curve.

Self-loop: 1-1 → 4 means draw a curve from dot 1 back to itself.

Counting Lines (Critical!)

Track how many lines each dot has:
- A dot with 3 lines is dead (cannot be used)
- A new dot starts with 2 lines (the curve passes through it)

Example Game (3 dots)

Initial state:

    1•           •2


         •3

Dot status: 1(0), 2(0), 3(0)

Message 1 (Alice):

Rules: https://strangerloops.com/experiments/pbem-games/sprouts.md

Move: 1-2 → 4

    1•----4•----•2


         •3

Dot status: 1(1), 2(1), 3(0), 4(2)

Your turn!

Message 2 (Bob):

Rules: https://strangerloops.com/experiments/pbem-games/sprouts.md

Move: 3-4 → 5

    1•----4•----•2
           |
           5•
           |
         •3

Dot status: 1(1), 2(1), 3(1), 4(3-DEAD), 5(2)

Your turn!

Message 3 (Alice):

Rules: https://strangerloops.com/experiments/pbem-games/sprouts.md

Move: 1-3 → 6

    1•----4•----•2
    |      |
    6•     5•
    |      |
    +----•3

Dot status: 1(2), 2(1), 3(2), 4(3-DEAD), 5(2), 6(2)

Your turn!

[Game continues until one player cannot move...]

ASCII Diagram Tips

• Use | for vertical lines
• Use - for horizontal lines
• Use / and \ for diagonals
• Use + for corners
• Mark dots with •[number] or just the number
• Note: ASCII is imperfect for curves; describe ambiguous cases in words

Mathematical Notes

• A game starting with n dots always ends after at least 2n and at most 3n-1 moves
• The game is finite — it always terminates
• For small n, optimal play is known:
• n=1: Second player wins
• n=2: Second player wins
• n=3: First player wins
• n=4: First player wins
• n=5: First player wins

Variant: Brussels Sprouts

Same rules, but:
- Start with crosses instead of dots (+)
- Each arm of the cross counts as a potential connection point
- Ends when all connection points are used

Starting a Game

Rules: https://strangerloops.com/experiments/pbem-games/sprouts.md

Sprouts challenge!

How many dots? (3 is standard, 2 is quick, 4+ gets complex)

Starting with 3 dots:

    1•           •2


         •3

Dot status: 1(0), 2(0), 3(0)

You go first!

Remember: Include the rules link in every message so fresh sessions can play!