n change$-$ringing on $4$ bells, the only bell swaps that are allowed between changes are the swaps

a$,$ b$,$ c$,$ d given by

a $=(12)(34),$ b $=(23),$ c $=(34),$ and d $=(12).$

$($a$)$ A plain hunt on $4$ bells starts with a round $1234$ and then alternates between swap a and swap

b until it returns to a round. Determine the list of changes involved. How many changes are

there?

$($b$)$ An extent on n bells is a set of changes that involves each permutation of the n bells exactly

once. Plain Bob Minimus is an extent on $4$ bells that starts with a round $1234,$ repeats the

list of swaps a$,$ b$,$ a$,$ b$,$ a$,$ b$,$ a$,$ c three times, and ends with a round. Determine the list of changes

involved.

Does any bell go plain hunting $($sweeping back and forth from end to end$)?$