Extending Sicherman Dice to 100-cell Calculation Tables

TL;DR

This paper explores the extension of Sicherman dice to a 100-cell addition table, identifying multiple solutions and establishing a formula for the number of solutions based on divisors of the table size.

Contribution

It introduces a novel analysis of 100-cell addition tables, finding multiple solutions through algebraic and geometric methods, and proposes a general formula for solution count.

Findings

Identified seven solutions for the 100-cell table.

Established a correspondence between algebraic and geometric solutions.

Proposed a formula for the number of solutions based on divisors of n.

Abstract

This paper discusses a 100-cell calculation table for addition, C(i,j)=Ai+Bj,(1<=i,j<=10), addressing the special case where the 100 cells C(i,j) comprise continuous numbers 0 through 99. We consider what sequences {Ai,Bj,(1<=i,j<=10)} are needed to fulfill these conditions, and the number of possible combinations. We first find seven solutions through factorization of the generating function used for Sicherman dice. We next find seven solutions through geometric means. We demonstrate a correspondence between the seven solutions from factorization and the seven from geometry. Finally, we suggest that the number of solutions N for an nxn=n^2-cell calculation table is N=(p-2)(p-1)+1, where p is the number of divisors of n.