The descriptor sequence is a sequence of numbers in which the digits of each number describe the preceding number. The first number is 1. This number consists of one 1, so the second number is 11 (that is, one-one). This consists of two 1's, so the third term is 21. This consists of one 2 and one 1, so the fourth term is 1211. The first six numbers in the sequence are:
1, 11, 21, 1211, 111221, 312211.
Show that no digit greater than 3 ever occurs, and that the string 333 never occurs.
Assume that there are descriptors that contain 333. Find the first such descriptor. The preceding descriptor must necessarily be something which contains three 3s and three of some other digit, like 333222.
But this is a contradiction, because this is before the first descriptor which contains 333. Therefore, our initial assumption was wrong.
Proof that 333 is impossible
