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#1

Hamiltonian circuit for a 1x1x1 cube.

To make a 1x1x1 cube even more of a challange, one could acount for the orientation of the cube when it sits on the table.

For instance one could definde the position as solved when the white face is on top and the green face is showing towards the observer

(F : green; U: white).

With this you would have 24 distinct states of the cube. Gods number is 2 in htm and it is 3 in qtm.

A path through all possible states of the cube that starts and end in the solved position (hamiltonian circuit, devil's algorithm) is this one:

FRU BLD FRU BLU FRU BLD FRU BLU

https://alg.cubing.net/?puzzle=1x1x1&alg=FRU_BLD%0AFRU_BLU%0AFRU_BLD%0AFRU_BLU

and this is another one:

UUUF UUUF' UUUF' UUUF UUUF U'U'U'F

https://alg.cubing.net/?puzzle=1x1x1&alg=UUU%0AF______%2F%2F_move_blue_up_%0AUUU%0AF-_____%2F%2F_move_orange_up%0AUUU%0AF-_____%2F%2F_move_yellow_up%0AUUU%0AF_____%2F%2F__move_green_up%0AUUU%0AF_____%2F%2F__move_red_up%0AU-U-U-%0AF____%2F%2F___back_to_start

open questions: how many different "hamiltonian circuits" do exist for the 1x1x1 cube ?

To make a 1x1x1 cube even more of a challange, one could acount for the orientation of the cube when it sits on the table.

For instance one could definde the position as solved when the white face is on top and the green face is showing towards the observer

(F : green; U: white).

With this you would have 24 distinct states of the cube. Gods number is 2 in htm and it is 3 in qtm.

A path through all possible states of the cube that starts and end in the solved position (hamiltonian circuit, devil's algorithm) is this one:

FRU BLD FRU BLU FRU BLD FRU BLU

https://alg.cubing.net/?puzzle=1x1x1&alg=FRU_BLD%0AFRU_BLU%0AFRU_BLD%0AFRU_BLU

and this is another one:

UUUF UUUF' UUUF' UUUF UUUF U'U'U'F

https://alg.cubing.net/?puzzle=1x1x1&alg=UUU%0AF______%2F%2F_move_blue_up_%0AUUU%0AF-_____%2F%2F_move_orange_up%0AUUU%0AF-_____%2F%2F_move_yellow_up%0AUUU%0AF_____%2F%2F__move_green_up%0AUUU%0AF_____%2F%2F__move_red_up%0AU-U-U-%0AF____%2F%2F___back_to_start

open questions: how many different "hamiltonian circuits" do exist for the 1x1x1 cube ?