[Paper Review] Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9

Inder J. Taneja|arXiv (Cornell University)|Feb 6, 2013

AI-based Problem Solving and Planning1 references3 citations

TL;DR

This paper presents a systematic exploration of writing all natural numbers from 0 to 11111 using the digits 1 through 9 exactly once in both increasing and decreasing sequential order. By employing basic arithmetic operations—addition, subtraction, multiplication, division, exponentiation—and parentheses, the author constructs expressions for nearly all numbers, with the notable exception of 10958 in the increasing sequence. The work extends prior efforts by incorporating subtraction and division to resolve gaps in earlier representations.

ABSTRACT

Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1 to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation, and division. In both the situations there are no missing numbers, except one, i.e., 10958 in the increasing case.

Motivation & Objective

• To represent all natural numbers from 0 to 11111 using digits 1 through 9 in strictly increasing and decreasing sequential order.
• To resolve gaps in prior works that missed approximately 1,250 numbers by including subtraction and division operations.
• To identify and document the sole unresolved number, 10958, in the increasing order representation.
• To explore the combinatorial explosion of possible expressions when extending operations beyond addition, multiplication, and exponentiation.
• To provide a comprehensive reference of sequential representations using digit sequences and standard arithmetic operations.

Proposed method

• The author uses all digits 1 to 9 exactly once in increasing or decreasing order to form expressions.
• Elementary operations include addition, subtraction, multiplication, division, exponentiation, and parentheses for grouping.
• Expressions are constructed through recursive and combinatorial exploration of digit concatenation and operation application.
• The author leverages computational scripts, assisted by T.J. Eckman, to verify and discover complex representations.
• Subtraction and division are systematically applied to resolve missing numbers from earlier works.
• The method is based on generating all valid combinations of operations and digit groupings under sequential constraints.

Experimental results

Research questions

• RQ1 Can all natural numbers from 0 to 11111 be represented using digits 1 to 9 in increasing order with basic arithmetic operations and parentheses?
• RQ2 Why is 10958 the only number that cannot be represented in increasing order under the given constraints?
• RQ3 How do the inclusion of subtraction and division affect the completeness of sequential number representations?
• RQ4 What is the combinatorial complexity of generating valid expressions from digit sequences under operation constraints?
• RQ5 To what extent can factorial, square root, or decimal operations resolve the missing representation of 10958?

Key findings

• All numbers from 0 to 11111 are successfully represented in both increasing and decreasing orders of digits 1 to 9, using +, −, ×, ÷, ^, and parentheses.
• Only one number, 10958, remains unrepresented in the increasing order case, despite extensive use of subtraction and division.
• The inclusion of subtraction and division resolved approximately 1,250 missing numbers from prior works that used only addition, multiplication, and exponentiation.
• A total of 1,256 numbers required subtraction and/or division—611 in increasing order and 645 in decreasing order—with 178 numbers common to both.
• The paper provides explicit expressions for all numbers from 0 to 11111, including complex cases like 11111 = 1 × 23 × 456 + 7 × 89 in increasing order.
• The representation of 11111 in decreasing order is −9 + 8 × 7 −6 × (5 −432 × 1), demonstrating the power of negative and nested operations.