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Non-Geometric Switch Toggling Game | Discovering Rules in Switch Systems

Dive into the strategic world of switch toggling games, exploring the rules and transitions in systems of switches. Understand the interplay between toggles, states, and parity for successful transitions.

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Non-Geometric Switch Toggling Game | Discovering Rules in Switch Systems

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  1. A Non-geometric Switch Toggling Game Megan Duke Muskingum University

  2. Lights Out by Tiger Relies on the position of the chosen switch

  3. An example of a system of 7 switches with a 5-toggle transition rule applied

  4. Graph of a system of 7 switches with a 5-toggle transition rule applied

  5. Graph of a system of 9 switches with a 4-toggle transition rule applied States and are in different components of the graph.

  6. Achieving the state depends on the parity of , the number of switches in the system and , the number of switches being toggled. If is odd, a system of switches can be transitioned from to . If is even, a system of switches can be transitioned from to only when is even.

  7. When is odd and Method: Apply to to get Apply to times to get Apply to to get decreases the number of switches on by

  8. When is odd and Method: Apply to to get Apply to to get Apply to to get Case: is odd This case is always done in steps.

  9. A system of switches with a -toggle transition rule applied A system of switches with a -toggle transition rule applied

  10. When is even and is odd Given is even, is even. From an initial state , any sequence of transition rules yielding the state will have . There are no transition rules to go from to .

  11. When is even and Method: Apply to to get Apply to times to get Apply to to get Consider only when is even decreases the number of switches on by

  12. Transitioning from a given initial state to a specified terminal state depends on the parity of . If is odd, a system of switches can be transitioned from to. If is even, a system of switches can be transitioned from toonly when .

  13. References

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