$[$ [Arithmetic. Mental calculation, etc.] $]$

Authors: Gaityerri G.A., Grierenko D.:

2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 and 9999 years)?
Let the current year be a palindrome of the form $\overline{a b b a}$. When will the next such year occur? Consider two cases:

a) $b=9$ (year of the form $\overline{a 99 a}, a<9)$. Then, 11 years later, another palindromic year will occur: $\overline{(a+1) 00(a+1)}$. For example, the years 3993 and 4004.

b) $b<9$. In this case, the next palindromic year will occur 110 years later: $\overline{a(b+1)(b+1) a}$. For example, the years 9339 and 9449. Therefore, the maximum number of consecutive non-palindromic years is 109.

## Answer

109 years.