Can one get to Room 11?
After briefly exploring the maze, one can see it is easy to get to the primes
2, 3, 5, 7, 13, 17, 19, 23, 29, 31, and 37. But strangely, the number 11 is
not in this list. The rules prevent one from going directly from 3 =
112 to 11 = 10112, since this move adds more than one
digit to the number. However, the rules do not prevent reaching Room 11 from
Room 43 = 1010112, Room 139 = 100010112, Room 523 =
10000010112, or even Room 32779 = 10000000000010112.
Changing the leading 1 to a 0 on one of these primes allows one to drop down
to Room 11 from above. However, there doesn't seem to be any way of reaching
these 4 rooms either. Can one prove that Room 11 is impossible to reach?
Actually, there is a simple proof that one cannot get to Room 11 in any number
of steps. Those who wish to try their hand at the proof can do so before
looking at the solution. Here's a hint: Of the
9592 primes less than 100000, about half of them (the exact number is unknown)
are accessible from Room 2.
Back to the
maze.