A Non-geometric Switch Toggling Game

Megan DukeMuskingum University

Lights Out by

TigerRelies on the position of the chosen switch

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

(7,0 )𝜏5,0

(2,5 )

(5,2 )

(0,7 )𝜏5,0

𝜏1,4

0 , 7 1 , 6 2 , 5 3 , 4

4 , 3 5 , 2 6 , 1 7 , 0

𝜏5,0𝜏5,0

𝜏1,4

Graph of a system of 7switches with a 5-toggle transition ruleapplied

0 , 9 1 , 8

2 , 7 3 , 6 4 , 5

5 , 4 6 , 3

7 , 2 8 , 1 9 , 0

Graph of a system of 9switches with a 4-toggle transition ruleapplied

States and are in different components of the graph.

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.

When is odd and

𝑘=2𝑚+1

decreases the number of switches on by

𝑛=2𝑘+𝑟Method:1. Apply to to get

2. Apply to times to get

3. Apply to to get

𝒏=𝟏𝟎 ,𝒌=𝟑

0 , 1 0 1 , 9

2 , 8 3 , 7 4 , 6 5 , 5 6 , 4

7 , 3 8 , 2 9 , 1 1 0 , 0

(7,3 )(6,4 )(5,5 )(4,6 )(3,7 )

} 𝜏2 ,14 times

(10,0 )𝜏3,0

(0,10 )𝜏3,0

When is odd and

This case is always done in steps.

𝑛−𝑘=2𝑎

Method:1. Apply to to get

2. Apply to to get

3. Apply to to get

Case: is odd

0 , 1 0 1 , 9 2 , 8 3 , 7 4 , 6 5 , 5

6 , 4 7 , 3 8 , 2 9 , 1 1 0 , 0

0 , 8 1 , 7 2 , 6 3 , 5

4 , 4 5 , 3

6 , 2 7 , 1 8 , 0 A system of switches with a -toggle transition rule applied

A system of switches with a -toggle transition rule applied

When is even and is odd (𝑥 , 𝑦 )𝜏𝑘−𝑤 ,𝑤

→(𝑥−𝑘+2𝑤 , 𝑦+𝑘−2𝑤 )

Given is even, is even.

From an initial state , any sequence of transition rules yielding the state will have .

𝑛≢0mod 2There are no transition rules to go from to .

When is even and

𝑘=2𝑚decreases the number of switches on by

𝑛=2𝑘+2𝑟

Method:1. Apply to to get

2. Apply to times to get

3. Apply to to get

Consider only when is even

0 , 1 4 1 , 1 3 2 , 1 2 3 , 1 1 4 , 1 0 5 , 9 6 , 8

7 , 7 8 , 6

9 , 5 1 0 , 4 1 1 , 3 1 2 , 2 1 3 , 1 1 4 , 0 (10 ,4 )(8 ,6 )(6 ,8 )(4,10 )

} 𝜏3 ,13 times

(14 ,0 )𝜏4 ,0

(0,1 4 )𝜏4 ,0

𝒏=𝟏𝟒 ,𝒌=𝟒

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 .

References