Problem 284Published on Saturday, 27th March 2010, 01:00 am; Solved by 818; Difficulty rating: 55%
The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 3762 = 141376. Let's call a number with this property a steady square.
Steady squares can also be observed in other numbering systems. In the base 14 numbering system, the 3-digit number c37 is also a steady square: c372 = aa0c37, and the sum of its digits is c+3+7=18 in the same numbering system. The letters a, b, c and d are used for the 10, 11, 12 and 13 digits respectively, in a manner similar to the hexadecimal numbering system.
For 1 ≤ n ≤ 9, the sum of the digits of all the n-digit steady squares in the base 14 numbering system is 2d8 (582 decimal). Steady squares with leading 0's are not allowed.
Find the sum of the digits of all the n-digit steady squares in the base 14 numbering system for
1 ≤ n ≤ 10000 (decimal) and give your answer in the base 14 system using lower case letters where necessary.