How many times are the hands perpendicular to each other strictly between the times on clock A (Monday night) and clock B (early Tuesday)?

A shows 9:00 p.m. and B shows 3:00 a.m., 6 hours apart. Right angles occur every 360/11 = 32 8/11 min (the relative angle grows 5.5° a minute and hits 90° or 270° every 180°). 6 h = 360 min = 11 such gaps. Both 9:00 and 3:00 are themselves right angles, so strictly between there are 11 − 1 = 10. (12 assumes two per hour; 11 counts one end.)