[ Inequalities. Case Analysis ]

A magical moment is when the number of minutes on a digital clock matches the number of hours. To brew a magical potion, it must be put on the fire and taken off the fire at magical moments. To make it taste good, it must be cooked for 1.5 to 2 hours. How long does it take to cook a delicious magical potion?

#
The number of hours on digital clocks equals the number of minutes 23 times a day: 00:00, 01:01, 02:02, .., 21:21, 22:22, 23:23.

If we are within one day, the difference between these numbers is either 1 hour 1 minute, or 2 hours 2 minutes, and so on. That is, if the potion is brewed without crossing midnight, it cannot be tasty. When crossing midnight, there are two ways to get a time difference of between one and a half to two hours: start at 22:22 and finish at 00:00, or start at 23:23 and finish at 01:01. In both cases, the potion needs to be brewed for 1 hour and 38 minutes.

## Answer

1 hour 38 minutes.