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 a digit greater than 3 is possible. Find the first number which contains a digit greater than 3. The preceding number, which it describes, must necessarily have the same digit repeated more than three times. For instance, 1111 or 22222. But this is not the way the descriptor works. Instead of 1-one and 1-one, the descriptor would be 2-ones. Therefore our initial assumption is wrong, and no digit greater than 3 is possible.
Proof that the digits greater than 3 are impossible
