Question

The numbers 828 and 313 are 3-digit palindromes where 828-313=515 which is also a palindrome. How many pairs (a, b) of 3-digit palindromes are there with a>b and with a-b also a 3-digit palindrome?

My Answer: 1980

Reasoning

Conditions

1. For palindromes aba and cdc to for a palindrome, then b>d.

2. To form a 3-digit palindrome, then a>1.

So we check a case by case analysis, starting with a=9 and varying b \(\displaystyle (0\leq{b}\leq{9})\)

Number: 909 919 929 939 .... 999

Pairs: 8 16 24 32 .... 80

Moving onto a=8

Number: 808 818 828 ..... 898

Pairs: 7 14 21 ..... 70

And so on to a=2

Number: 202 212 .... 292

Pairs: 1 2 .... 10

So we a pattern of

8x(1+2+3+...+10) + 7x(1+2+3+...+10) + 1x(1+2+3+....10)

This becomes

(1+2+3+...+10)(1+2+3+...+8)=55x36

Ans=1980

So could someone please check this.

Thanks