## Merlin's Machine

Welldone!

Reset

Welldone!

Reset

The rules of Merlin's Machine are as follows:

A 3 x 3 grid of touch buttons that light up are started in a random sequence.

When a button is pressed then itself and the neighbouring lights are switch on/off, accordingly.

The diagram shows the buttons labelled, for example, row 1 column 1 (r1 c1) as 11.

To solve I've used the following definitions:

Set X = {1,2,3}, Set S = {0,1}.

The cross product Ω = X x X, ie Ω = {(1,1),(1,2),(1,3),....,(3,3)}

Mapping α : n ↔ t⋅3+n-3, (t,n) ∈ Ω

Set C = {x(α) → c ∈ S}, eg C = {1,0,1,0,1,1,0,0,1}, the machines current state.

Binary operations (s,m),(t,n) ∈ Ω, Ω ~ Ω = (s+t-2,m+n-2),

c^{*}(ω) if c(ω) = 0 then c(ω) = 1 else c(ω) = 0

The algorithm goes a bit like this:

[loop till game complete]

Pressed ω ∈ Ω,

∀ Ω if ω ~ Ω then c^{*}(ω)

∀ c ∈ C if c = 0 then [game complete]

[repeat]

To get random C then generate random r (0<r<512) such that

∀ c ∈ C, c → bitwise r.

Any Q's press the big envelope below and give me a shout.

A 3 x 3 grid of touch buttons that light up are started in a random sequence.

When a button is pressed then itself and the neighbouring lights are switch on/off, accordingly.

The diagram shows the buttons labelled, for example, row 1 column 1 (r1 c1) as 11.

To solve I've used the following definitions:

Set X = {1,2,3}, Set S = {0,1}.

The cross product Ω = X x X, ie Ω = {(1,1),(1,2),(1,3),....,(3,3)}

Mapping α : n ↔ t⋅3+n-3, (t,n) ∈ Ω

Set C = {x(α) → c ∈ S}, eg C = {1,0,1,0,1,1,0,0,1}, the machines current state.

Binary operations (s,m),(t,n) ∈ Ω, Ω ~ Ω = (s+t-2,m+n-2),

c

The algorithm goes a bit like this:

[loop till game complete]

Pressed ω ∈ Ω,

∀ Ω if ω ~ Ω then c

∀ c ∈ C if c = 0 then [game complete]

[repeat]

To get random C then generate random r (0<r<512) such that

∀ c ∈ C, c → bitwise r.

Any Q's press the big envelope below and give me a shout.

Wikis:

Merlin's Game

Adventures in Group Theory

Professor David Joyner