N is a m-th power of an integer.
It consists of n distinct digits, averaging n.

Find n,m and N, using your head + pen and paper only.

Check, whether there exist additional solutions (computer allowed).

Clearly, n>1; m>1 - to exclude trivial cases.

I found it handy to have tables of exponents and squares on the wall in my classroom.
Looking up there counts as pen and paper IMO.
I see a few three digit examples:
6^3 = 216
12^2 = 144
15^2 = 225
18^2 = 324
21^2 = 441
in each case the sum of digits is 9 and so average 3.

There are obviously no 2 digit examples.
There could be other 3 and 4+ digits examples.

Edit:   I thought of another: 30^2 = 900

Edited on January 27, 2014, 12:44 pm

Posted by Jer on 2014-01-27 12:42:36