A sequence {S(x)}, each of whose terms is a positive integer, is defined as:

S(x) = x/2, if x is even, and:

S(x) = x+3, if x is odd

Given that n is an odd positive integer and, S(S(S(n))) = 27, determine sod(n).

Note: sod(n) denotes the sum of the base-10 digits of n.

A little background information on this sequence, none of which is necessary to solve the problem:

No matter what the value of x, the sequence seems to wind up repeating (cycling) through the digits 4,2,1.  For instance, if x = 7, the sequence is

7,10,5,8,4,2,1,4,2,1,4,2,1,4,...

Last time I checked (years ago) it was generally believed that this was true for all x, but that it could not be proved that it was true for all x.

(Godel has proved that there are some theorems that are true but unprovable in any sufficiently rich mathematical system, and many believed that this was one of them.)

Of course, it was also widely believed that the 4-color map theorem was unprovable, until that theorem was proved.

Posted by Steve Herman on 2009-03-02 11:05:47