## Palindromic Years

#### Problem

A palindromic year is one which reads the same forwards and backwards. There were no other palindromic years between 1991 and 2002, which is a gap of 11 years.

Since year 1 A.D., what was the largest gap between two consecutive palindromic years?

#### Solution

Changing from 1-digit to 2-digit: 11 9 = 2; 2-digit to 3-digit: 101 99 = 2; and 3-digit to 4-digit: 1001 999 = 2.

That is, the difference is always 2 years.

For 2-digit palindromic years, the gap will always be 11:

E.g. 77 66 = 11.

For 3-digit years, the gap will be 10 within the same century or 11 years when a century changes:

E.g. 686 676 = 10

E.g. 505 494 = 11.

For 4-digit years, the gap is 11 years as the millenium changes otherwise it will be 110 years as centuries change:

E.g. 2002 1991 = 11

E.g. 1441 1331 = 110.

As there is no other possible cases, the maximum gap between consecutive palindromic years is 110 and it happened 9 times.

In general, what is the largest gap you can have between two consecutive palindromic numbers?