A watch which gains uniformly is 2 minutes slow at 10AM on Wednesday and $$2\frac{1}{2}$$ minutes fast at 1 PM the following day. When will it show the correct time?

At 10 AM on Wednesday, the watch is 2 minutes slow. At 1 PM on Thursday, the watch is 2.5 minutes fast. Thus, the watch gains uniformly.

Total time interval from 10 AM Wednesday to 1 PM Thursday = 27 hours. 

Total gain = 2 (to cover slow) + 2.5 (extra fast) = 4.5 minutes in 27 hours.

Thus, Gain per hour = 4.5/27 minutes = 1/6 minutes

Now, at 10 AM Wednesday, it is 2 minutes slow. To display the correct time, it must gain these 2 minutes; therefore, we need to determine the time required to gain these 2 minutes. Each hour, the clock is gaining 1/6 minutes, thus - 

Time required to gain 2 minutes = 2 / (1/6) = 12 hours.

12 hours after 10 AM Wednesday will be = 10 AM Wednesday + 12 hours = 10 PM Wednesday.