Answer:  If you're like me you immediately assumed it had to be between 11 and 12. But after adding these with the 3 quarter hour chimes you still came up with only 26.  Stumped?  Consider this solution:  The clock will chime 12 two times within an hour if the owner sets the clock back one hour just after midnight on the night people adjust their clocks back from daylight savings time to standard time.