<p>If the algebra doesn’t come to mind, this problem can be solved by brute force. Make a table of “real time” and “Bad Clock time” and keep adding rows until the two times match. Then look to see how much real time has passed.</p>
<p>RT 12 BCT 12
RT 1 BCT 12:20
RT 2 BCT 12:40
RT 3 BCT 1
RT 4 BCT 1:20
RT 5 BCT 1:40
RT 6 BCT 2
RT 7 BCT 2:20
RT 8 BCT 2:40
RT 9 BCT 3
RT 10 BCT 3:20
RT 11 BCT 3:40
RT 12 midnight BCT 4
RT 1 BCT 4:20
RT 2 BCT 4:40
RT 3 BCT 5
RT 4 BCT 5:20
RT 5 BCT 5:40
RT 6 BCT 6 </p>
<p>There you go. After 18 hours the Bad Clock shows the right time.</p>