How many times do the hands form an angle of exactly 60° strictly between 1 p.m. and 7 p.m. of the same afternoon?

The minute hand gains 5.5° a minute on the hour hand, so 60° angles come at T = (60 + 360k) ÷ 5.5 or (300 + 360k) ÷ 5.5 minutes after 12. Between 1:00 and 7:00 these are about 1:16, 2:00, 2:22, 3:05, 3:27, 4:11, 4:33, 5:16, 5:38, 6:22 and 6:44: 11 times. (12 assumes exactly two per hour, but in 12 hours there are only 22, and the hour from 1 to 2 holds just one.)