Partition Problems and a Pattern of Vertical Sums

TL;DR

This paper analyzes a partition problem involving three subsets of non-negative integers, revealing a standard solution pattern based on modular arithmetic, and explores the classification of similar partition statements.

Contribution

It introduces a new algorithm inspired by alternating differences, classifies all similar statements, and constructs infinitely many partitions equivalent to the standard one.

Findings

Identifies the standard partition based on modulo five remainders.

Classifies 279,936 similar partition statements and their resulting partitions.

Constructs infinitely many partitions equivalent to the standard partition.

Abstract

We give a possible explanation for the mystery of a missing number in the statement of a problem that asks for the non-negative integers to be partitioned into three subsets. We interpret the missing number as one of the clues that can lead to a more standard solution to the problem, using only congruence modulo five, and we give the details to the new solution, which is based on an algorithm inspired by noticing alternating differences between sums of elements of the same rank in the three sets. Our new solution is equivalent to the partition consisting of numbers with remainders one or three modulo five, two or four modulo five, and multiples of five, which we call the standard partition. We then find all other similar statements with the same pattern of sums, we apply the algorithm to them, and we describe all the partitions obtained, up to a certain equivalence. There are $279936$ different such statements, they produce twenty different partitions (other than the standard one) whose sets of the first five columns are not permutations of each other, and only one of them (the one produced by the original statement of the problem we study) is equivalent to the standard partition. Finally, we construct infinitely many partitions equivalent to the standard one, and we give a possible generalization and a sample partition problem asking for the non-negative integers to be partitioned into four sets.