<pre>
The hands overlap every 12/11 hour.  So here are the times that they
overlap:

<pre>
 1.  midnight
 2.  1 + 1/11 am
 3.  2 + 2/11 am
 4.  3 + 3/11 am
 5.  4 + 4/11 am
 6.  5 + 5/11 am
 7.  6 + 6/11 am
 8.  7 + 7/11 am
 9.  8 + 8/11 am
 10. 9 + 9/11 am
 11. 10 + 10/11 am
 12. noon
 13.  1 + 1/11 pm
 14.  2 + 2/11 pm
 15.  3 + 3/11 pm
 16.  4 + 4/11 pm
 17.  5 + 5/11 pm
 18.  6 + 6/11 pm
 19.  7 + 7/11 pm
 20.  8 + 8/11 pm
 21. 9 + 9/11 pm
 22. 10 + 10/11 pm

 If midnight is considered part of both days, then the answer is 23 times.
</pre>
Otherwise the answer is 22 times.
</pre>
