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Work Shown:
Let a,b,c be three different two-digit palindromes
So N = a+b+c is the sum of those values
The two digit palindromes are {11,22,33,44,55,66,77,88,99} which will represent the possible values of a,b,c.
Let's pick the three smallest values and add them up
11+22+33 = 66
The result isn't three digits, so let's erase 11 and pick 44 this time
22+33+44 = 99
which also isn't three digits. Bump off the smallest value and pick the next largest two digit palindrome, add them up to get
33+44+55 = 132
This sum is now three digits and it's also not a palindrome
The sum of the digits of 132 is 1+3+2 = 6 which is the answer.
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So in short, the plan to solve this is to pick the smallest items in the set
{11,22,33,44,55,66,77,88,99}
and use trial and error to see which items add to a three digit non-palindromic number. We picked the smallest items to ensure that N is the smallest three digit number.