If the clock gains a minute a day, then it will take 720 days to gain 12 hours. On this assumption you need to work out the day and date for 720 days after 1 June 2015. This assumption of gaining exactly a minute a day employs a principle similar to calculating simple interest.
However, if the clock is gaining at the same rate every day, its mechanism is working at 1441/1440 compared to the other clock. 1441/1440=1.00069444..., so the clock is actually gaining at a rate slightly more than a minute a day. After one day, it gains exactly a minute, because when the rate is applied to 1440 minutes, the result is exactly 1441, or 1 minute over a day. For example, by 3 June 2015, 2 days after sync, it gains about 2.000694 minutes, not 2 minutes; after a month it gains about 30.304 minutes; after a year it gains about 415.553 minutes or 6.94255 hours, rather than 6.1 hours. Instead of taking 720 days to sync (gain exactly 12 hours) it takes considerably fewer days (about 584). The problem is similar to calculating compound interest.