I'm reading the chapter "Fixed Point Puzzles" and there is a problem titled "An Open Problem" after problem #18. The chapter introduces a machine which operates on strings of upper case letters, which are then processed according to some internal rules, and a new string is output. The rules are as follows (so far anyway):
Given a input string, let's call it x, the output is a string, lets call it y. x->y means x outputs y.
RULE Q: Given string x, the output of Qx is x. Qx->x
RULE C: Given string x which outputs string y, x->y, Cx outputs yQy: Cx-yQy
RULE R: Given string x which outputs string y, Rx outputs yy. Rx->yy
RULE V: Given string x, Vx outputs the reverse of y.
Now the problem states: Is there such a string x that the output is xA, where A is simply a letter? Can we prove this to be true or false?
My question is, does this problem have a name? Has it been solved? I do believe I have shown it to be false, but perhaps it is a simple problem >_>