Crazy Sequential Representations of Numbers for Small Bases

TL;DR

This paper generalizes the concept of crazy sequential representations of numbers from base ten to arbitrary bases, analyzing the ranges, algorithms, and patterns across bases three to ten, and discusses related mathematical and complexity problems.

Contribution

It extends previous work on sequential number representations to arbitrary bases and explores the associated combinatorial, algorithmic, and pattern-related challenges.

Findings

Analyzed ranges of representable numbers across bases 3 to 10

Developed efficient algorithms for finding sequential representations

Identified interesting patterns and open problems in the area

Abstract

Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers- equations made of sequences of one through nine (or nine through one) that evaluate to a number. All previous work on this type of puzzle has focused only on base ten numbers and whether a solution existed. We generalize this concept and examine how this extends to arbitrary bases, the ranges of possible numbers, the combinatorial challenge of finding the numbers, efficient algorithms, and some interesting patterns across any base. For the analysis, we focus on bases three through ten. Further, we outline several interesting mathematical and algorithmic complexity problems related to this area that have yet to be considered.