I am a four digit number that is the same when my digits are written backwards.

Implies first and last digit is same and 2nd and 3rd digit is same.

let first digit be x ,hence last digit will also x

Let 2nd digit=3rd digit=y

The sum of my outside digits is 6 times the sum of my two middle digits

hence,2x=12y

x=6y

if, y=1,than x=6

so number will be 6116

if y=2,then x=12 which is not possible.

Hence the four digit number is 6116