A Non-geometric Switch Toggling Game

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

9. 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.

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

11. 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 .

12. 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

14. 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 .

15. References