Ed Pegg's challenge (see Maze of Life) was successfully solved in June 2003 by Berend Jan van der Zwaag who discovered the maximum score is indeed infinity, by virtue of offering a loop that visits the end state and then returns to the start state.
To explore the loop in action use pre-loaded applet below.
Hit undo-all and then redo to step forwards from the original start state, first to the loop entry state (illustrated) and then on round the loop back to the entry state. Note the blue cell visits both the original start and goal points (illustrated in red) enroute.
The diagram on the right illustrates the loop entry state and original start/end points (red). The loop visits both red squares and returns to the entry state.
Start as follows:
Then, as often as you like, visit goal and starting point and return to the exact same cell-arrangement following the looping string:
>>SL\SSRRLUSSSDSDRU<LS/< SSSSSSS/R\U/S><UDS/R>\RL< SDL<DSSS/\UD.
applet - © Andrea Gilbert 2000-2002
applet JS conversion - cheerpj transpiler from Leaning Technologies - 2020