Step 1: Find the number of days between 2nd June and 3rd July.
June has 30 days, so from the 2nd of June to the 2nd of July is exactly 30 days.
Adding one more day to reach the 3rd of July gives 30 + 1 = 31 days in total from the 2nd of June to the 3rd of July.
Step 2: Convert the day gap into full weeks and extra days.
A week has 7 days, so divide 31 by 7: 31 = 7 x 4 + 3.
This means 31 days is exactly 4 full weeks plus 3 extra days.
Step 3: Use the extra days to shift the weekday.
Whole weeks do not change the day of the week, since the same weekday repeats every 7 days. So only the leftover 3 days matter.
Starting from Thursday (2nd June) and moving forward 3 days: Thursday + 1 day = Friday, Friday + 1 day = Saturday, Saturday + 1 day = Sunday.
Step 4: State the day for 3rd July.
After moving 3 days forward from Thursday, we land on Sunday.
Final Answer:
The 3rd of July falls on a Sunday.
\[ \boxed{\text{Sunday}} \]