$5.$

Let$$s extend the jug filling scenario of Section $9\mathrm{.}1\mathrm{.}3$ to three jugs and a receptacle.

Suppose the jugs can hold a$,$ b and c gallons of water, respectively.

The receptacle can be used to store an unlimited amount of water, but has no

measurement markings. Excess water can be dumped into the drain. Among the

possible moves$1.$ fill a bucket from the hose,

$2.$ pour from the receptacle to a bucket until the bucket is full or the receptacle

is empty, whichever happens first,

$3.$ empty a bucket to the drain,

$4.$ empty a bucket to the receptacle, and

$5.$ pour from one bucket to another until either the first is empty or the second

is full.

$($a$)$ Model this scenario with a state machine. $($What are the states? How does a

state change in response to a move?$)$

$($b$)$ Prove that Bruce can get k $2$ N gallons of water into the receptacle using the

above operations if gcd$.$a; b; c$/$ j k$.$