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

overlap

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.

Otherwise the answer is 22 times.

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