Math’s “look-and-say” sequence goes: 1, 11, 21, 1211, 111221, 312211, 13112221, ... Each entry describes the one before it. We start with 1, then observe “we have one 1” and encode that as the next entry: 11. Now we have two 1s, encoded as 21. That in turn contains one 2 and one 1: 1211, and so forth.
Which digits from 1 to 9 never appear in any number in the look-and-say sequence?
You can start similar sequences with positive integers other than 1. For example, starting with 4445 yields the sequence: 4445, 3415, 13141115, ..., and so on. All such sequences, regardless of the starting number, eventually grow to infinity, except for one. Which starting number gives rise to the exception?
For some starting numbers, the sequence decreases before it increases. What is the smallest starting number such that the second entry in the sequence is smaller than the first and the third entry in the sequence is smaller than the second?
Only the digits 1, 2 and 3 ever appear in the original look-and-say sequence. To see why, notice that in every term of the sequence, the digits alternate between those that count the lengths of runs of other digits and those that get counted. For example, the fifth term is 312211. We’ll underline the digits that were counted from the previous term: 312211. The previous term had a run of 1s followed by a run of 2s followed by a run of 1s. The lengths of those runs are described by the digits that are not underlined. Notice that it wouldn’t make sense to say the previous term had a run of 1s followed by a run of 2s followed by another run of 2s, because two consecutive runs of the same digit would just get collapsed into a single count.
In order for the digit 4 or higher to first appear, the previous term must have a run of four or more of the same digit. As an example, let’s say a term has four 1s in a row: ...1111... Because digits that count and digits that get counted alternate, two of these 1s must be getting counted. In other words, the previous term must have had a run of 1s followed by another run of 1s, which we just established doesn’t make sense.
The starting number 22 leads to a sequence that never grows. It is described as “two 2s,” so the second entry in the sequence would be 22 again. This continues forever. It’s the only starting number that doesn’t eventually grow to infinity. This was proven by the legendary mathematician John Conway.
The smallest number that leads to two consecutive decreases is 333: the sequence is 333, 33 and 23, and then it increases to 1213.
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