Starting from an intitial sequence of ♥'s and ♠'s, get to the goal sequence in a certain number of moves. To do this you can use 3 different replacement rules, one at a time, on any consecutive substring of the present sequence. You may not have more than 6 symbols at any one time.

Rule #1: ♥♠ → ♠♥♥ Rule #2: ♥ → ♥♠ Rule #3: ♥♠♠ → ♥♠ Goal: ♥♥ → ♠♥♥♥♥♥ in 6 moves |
Rule #1: ♥ → ♠♥♠ Rule #2: ♠♥ → ♠♠ Rule #3: ♠♥♠ → ♥ Goal: ♥♥♠ → ♥ in 6 moves |

Rule #1: ♠♠ → ♥♠♥ Rule #2: ♥ → ♠♠♠ Rule #3: ♥♠♠ → ♠♥ Goal: ♠♥♠ → ♠♥♠♥♥ in 8 moves |
Rule #1: ♠ → ♥♠♠ Rule #2: ♥♠ → ♥ Rule #3: ♠ → ♠♠ Goal: ♥♠ → ♥♥♠♥♥ in 8 moves |

Rule #1: ♥♥ → ♠♠♠ Rule #2: ♥♥♠ → ♠♥ Rule #3: ♠ → ♥♥♥ Goal: ♥♠ → ♠♠♥♠♠ in 10 moves |
Rule #1: ♠ → ♥♥ Rule #2: ♥♠♥ → ♠ Rule #3 : ♥ → ♠♠♥ Goal: ♥ → ♠♥♠♥♥ in 10 moves |

Rule #1: ♠♠♥ → ♥♠ Rule #2: ♥♥♠ → ♠♠ Rule #3: ♥ → ♥♥♠ Goal: ♥♥♥ → ♠♥♥♠♠♠ in 12 moves |
Rule #1: ♠♥♠ → ♥ Rule #2: ♥ → ♠♥♥ Rule #3: ♥♥♠ → ♠♠ Goal: ♠♥♠ → ♥♠♠♥♥ in 12 moves |

The answers are here.