To determine the day on which Shashikanth was born, we need to consider the given information and use the concept of finding the day of the week for a given date.

Given information:
- Suresh was born on October 4, 1999.
- Shashikanth was born 6 days before Suresh.
- The Independence day of that year fell on Sunday.

First, let's find out the day of the week for Suresh's birthdate, October 4, 1999.

Step 1: Finding the number of days from the reference day
We will use the concept of finding the number of odd days to calculate the number of days from the reference day (Sunday) to October 4, 1999.

- Number of days in a non-leap year = 365
- Number of days in a leap year = 366

Since 2000 was a leap year, we have:
- Number of days from 2000 to 1999 = 1 day

Next, we need to find the number of days from January 1, 1999, to October 4, 1999.

- Number of days from January 1 to February 1 = 31 days
- Number of days from February 1 to March 1 = 28 days
- Number of days from March 1 to April 1 = 31 days
- Number of days from April 1 to May 1 = 30 days
- Number of days from May 1 to June 1 = 31 days
- Number of days from June 1 to July 1 = 30 days
- Number of days from July 1 to August 1 = 31 days
- Number of days from August 1 to September 1 = 31 days
- Number of days from September 1 to October 1 = 30 days
- Number of days from October 1 to October 4 = 4 days

Adding up all these days, we get:
31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 4 = 277 days

Step 2: Finding the number of odd days
To find the number of odd days, we divide the total number of days by 7 and take the remainder.

277 mod 7 = 1

So, there is 1 odd day from the reference day (Sunday) to Suresh's birthdate (October 4, 1999).

Step 3: Finding the day of the week
To find the day of the week for Suresh's birthdate, we count the odd day(s) from the reference day.

1 odd day from Sunday brings us to Monday. Therefore, Suresh was born on a Monday.

Now, to find the day of the week for Shashikanth's birthdate, we need to subtract 6 days from the day of the week for Suresh's birthdate (Monday).

Monday - 6 days = Tuesday

Therefore, Shashikanth was born on a Tuesday.