Having trouble constructing this sequential circuit
In a Moore machine each state is associated with a certain output or in your case a certain 6 outputs.
Look at what combinations of the 6 outputs are repeated. Where the outputs are the same in your photo, that combination of outputs can be represented by the same state in a Moore machine.
For example outputs of 000000 occur 3 times but can be represented by the same state. Outputs of 111111 also occurs 3 times and can also be represented by a single state. 000111 occurs twice but can be represented by a single state and 111000 also occurs twice but can be represented by a single state.
I count that reduces total number of states to 12.
In a Moore machine the inputs written on the transition arrows then dictate which is the next state that the sequence transitions to from the present state.