Mysterious phone numbers

A Viennese mathematician asked a young lady he knew, to give him a phone number, through which you could talk to her. The young lady, not really wishing for this conversation, replied jokingly, that in the office, where he works, there are four phones; in each telephone number all digits are different, but these four numbers have some common properties, namely: the sum of the digits of each number equals 10, when to each of these numbers we add a number written with the same digits, but in reverse order, then we will get four completely identical and equal-digit numbers.

— Niech panu to wystarczy — zakończyła, saying goodbye with a slightly malicious smile.

She was convinced, that of course with such vague guidelines no one will be able to "be wise."”. But it happened otherwise, and to the great surprise of the mischievous young lady, soon the voice of this boring gentleman spoke on one of her telephones..

How could he . . . guess those mysterious numbers?

Well, the mathematician knew, that all Viennese telephones have numbers between 20 000 a 99 999.

Suppose, that one of the office's phones has an ABCDE number, where subsequent numbers are marked with letters. According to the condition, the sum of the given number and the number with reversed order of digits is to be an "equal-digit" number:

ABCDE
+ EDCBA
FFFFF

And that is only possible then, when E + A = A + E = F, D + B = B + D = F i C + C = F.

Moreover, we know, that A + B + C + D + E = 10; from this conclusion, that F = 4 i C = 2.

The number A can be 3 or 4. Now it's easy to decipher four phone numbers:

30241, 34201, 41230, 43210.