Problem 473

Solution 1

Every 4-digit palindrome is of the form $abba$, where $a$ is a digit between 1 and 9 inclusive and $b$ is a digit between 0 and 9 inclusive (and $b$ is not necessarily different than $a$).

Every 5-digit palindrome is of the form $abcba$, where $a$ is a digit between 1 and 9 inclusive, $b$ is a digit between 0 and 9 inclusive (and $b$ is not necessarily different than $a$), and $c$ is a digit between 0 and 9 inclusive (and $c$ is not necessarily different than $a$ and $b$).

That is, for every 4-digit palindrome $abba$ there are 10 possible digits $c$ so that $abcba$ is a 5-digit palindrome.

For example if $a=2$ and $c=3$, then the 4-digit palindrome $2332$ can be used to create the 10 5-digit palindromes: $23032, 23132, 23232, 23332, 23432, 23532, 23632, 23732, 23832, 23932$.
Thus, for every 4-digit palindrome $abba$, there are exactly 10 5-digit palindromes $abcba$ and so the ratio of the number of 4-digit palindromes to the number of 5-digit palindromes is $1:10$.

Solution 2

Every 4-digit palindrome is of the form $abba$, where $a$ is a digit between 1 and 9 inclusive and $b$ is a digit between 0 and 9 inclusive (and $b$ is not necessarily different than $a$).

There are 9 choices for the first digit $a$ and, for each of these choices, there are 10 choices for the second digit $b$ or $9\times10=90$ choices for the first two digits $ab$.

Once the first two digits of the 4-digit palindrome are chosen, then the third and fourth digits are also determined (since the third digit must equal the second and the fourth must equal the first).

That is, there are 90 4-digit palindromes.

Every 5-digit palindrome is of the form $defed$, where $d$ is a digit from 1 to 9 inclusive and $e$ is a digit from 0 to 9 inclusive (and $e$ is not necessarily different than $d$) and $f$ is a digit from 0 to 9 inclusive (and $f$ is not necessarily different than $d$ and $e$).

There are 9 choices for the first digit $d$ and 10 choices for the second digit $e$ and 10 choices for the third digit $f$ or $9\times10\times10=900$ choices for the first three digits $def$.

Once the first three digits of the 5-digit palindrome are chosen, then the fourth and fifth digits are also determined (since the fourth digit must equal the second and the fifth must equal the first).

That is, there are 900 5-digit palindromes.
Thus, the ratio of the number of 4-digit palindromes to the number of 5-digit palindromes is $90:900$ or $1:10$.