The Moser's formula for the division of the circle by chords problem revisited

TL;DR

This paper revisits Moser's formula for dividing a circle with chords, reformulating the problem using a fourth order difference equation to derive the correct number of regions formed.

Contribution

It introduces a novel approach by applying a fourth order difference equation to derive Moser's formula, clarifying the combinatorial problem.

Findings

Derives Moser's formula using difference equations

Provides a new mathematical framework for circle division problems

Confirms the correctness of Moser's formula through this approach

Abstract

The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoning fails as was pointed out by Leo Moser in the Mathematical Miscellany in 1949. The formula that gives the right number of regions can be deduced by combinatorics reasoning using the Euler's planar graph formula, etc. My contribution in the present work is to reformulate and solve such problem in terms of a fourth order difference equation and to obtain the formula proposed by Leo Moser.