Palindromic Day 27572

Every 100 days another palindrome day rolls by and yesterday I celebrated palindromic day 27573. Now this palindrome has an arithmetic digital root that is equal to its central digit of 5. This is because:$$ 27572 \rightarrow 2+7+5+7+2= 23 \rightarrow 2 + 3 =5 $$However, the absolute difference between the first two digits (and of course the last two digits as well) is also equal to the digital root and the central digit.

| 2 - 7 | = 5 = | 7 - 2 |

This makes the palindrome extra special and in the range of five digit numbers from 10000 to 99999 only the following palindromes have the properties previously mentioned. These are (permalink):

18781, 27572, 36363, 45154, 54145, 63336, 72527, 81718, 90909

If we allowed leading zeros then we would have:

• 09990 has the same digits as 90909

| 0 - 9 | = 9 = | 9 - 0 |

• 18781 has the same digits as 81718

| 1 - 8 | = 7 = | 8 - 1 |

• 27572 has the same digits as 72527

| 2 - 7 | = 5 = | 7 - 2 |

• 36363 has the same digits as 63336

| 3 - 6 | = 3 = | 6 - 3 |

• 45154 has the same digits as 54145

| 4 - 5 | = 1 = | 5 - 4 |

Remember that the central digit is also the arithmetic digital root of the number.