## The Mathematics of Duelling

Duelling with pistols. If you were the one issuing the challenge, your dilemma was that custom dictated that your adversary be allowed to shoot first. Only then, if you were still able to shoot, would you be permitted to seek “satisfaction”.

How much of an advantage does the first shooter really have? In this article, we build a simple probability model, and implement a numerical model in a few lines of R code.

Two gentleman face off in the snow. Convention dictates the challenged shoots first.

## Finite Summation of Integer Powers (Part 3)

(Discrete Mathematics Techniques III)

Abstract
We find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem $S_p(N) = \sum_{k=1}^{N} k^p$. Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the $p+1$ independent monomials $a_j N^j$ ($1 \leq j \leq p+1$), where the $a_j$ are unknown coefficients. Using the recurrence relation $S_p(N+1) = S_p(N) + (N+1)^p$, we obtain a linear combination of the monomials, which reduces to an easily solvable $(p+1)$-by-$(p+1)$ triangular linear system in the unknown coefficients $a_j$ of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.

## Finite Summation of Integer Powers (Part 2)

(Discrete Mathematics Techniques II)

Abstract
We solve the general case of the finite-summation-of-integer-powers problem $S_p(N) = \sum_{k=1}^{N} k^p$ for arbitrary $p$, and obtain a $p$-th order recurrence relation that can be used to iteratively obtain the closed form polynomial for $S_p(N)$ for any given $p$. Source code is given for computing these polynomials using Maxima, an open-source (free) symbolic computation platform. (Note: This article generalizes the recurrence relation approach that is motivated and illustrated for small $p$ in Part 1. A direct matrix method for computing the closed form solutions is given in Part 3.)

## Sums of Powers (Part 1)

Abstract
This paper uses recurrence relations to find a closed form solution to the sum-of-powers problem $S_r(n) = \sum_{k=1}^{n} k^r$ for any given integer $r$. We use Maxima, a free symbolic computation package, to crunch through messy algebraic expressions and reach a simplified closed form. A solution to the general case (arbitrary $r$) is developed in Part 2. A matrix alternative to the general case solution is given in Part 3. Source code is provided for all solutions.

## Mathematics Toolset

…For industry or research.

Over the coming months, I’ll be posting articles as part of a series on setting up a toolset for Mathematics work in industry or research.

I’ll be emphasizing open source software. Though the primary target is the Windows PC platform (dominant in industry), I will list alternatives for Linux/Unix.

## Good mathematical technique and the case for mathematical insight

Good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, one can often pass from a single insight to the solution of a family of problems, and in some cases, to the solution of the general question itself. … Good mathematical technique has built within it the mathematical insight of the best of previous generations.