32containing

32containing is a term used to describe numbers whose decimal representation contains the consecutive digits 32 as a substring. The primary usage is in discussions of digit patterns and string-search problems. Examples of 32-containing numbers include 32, 132, 320, 5321, and 1232. Numbers such as 3012 or 4213 do not have the substring 32 and are not 32-containing. The term is informal and appears mainly in introductory explanations of digit patterns rather than in formal number theory.

Counting and analysis: For a given length n, the number of n-digit strings that are 32-containing can be found by subtracting from the total 10^n the number of length-n strings that avoid the substring 32. The avoidance set can be counted using a simple finite automaton that tracks whether the last digit was 3, enabling a recurrence relation to compute the count. As n grows, the proportion of 32-containing strings increases, and for sufficiently large n, it becomes common to encounter the substring 32 in random digit sequences.

Variants and extensions: The concept extends to other substrings and patterns, such as 42-containing or more general regular-expression-defined motifs. In computer science, checking whether a string is 32-containing is a canonical instance of substring search and can be implemented efficiently with standard algorithms or regular expressions.

See also: substring, pattern avoidance, regular expression, automaton.

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