Winning Strategy for Dice Game Farkle

The dice game Farkle uses 6 six-sided dice and the objective of the game is to be the first person to score 10,000 points. The dice game involves rolling against the odds and possibly lose your running total points. Generating functions are used to determine the probability of various roll patterns and their expected values of points earned given the number of dice rolled. We are assuming the maximum Farkle score is attained for each roll for computations. The probability of rolling a Farkle for each case and the expected values are used as guidelines for decision-making during game play.https://commons.und.edu/es-showcase/1000/thumbnail.jp

Groups Of Units Of Z_P[X] Modulo F(X)

University of Minnesota M.S. thesis. May 2020. Major: Mathematics. Advisor: Joseph Gallian. 1 computer file (PDF); iii, 37 pages.The set $\mathbb{Z}_p[x]$ consists of all polynomials with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. If a polynomial $f(x)$ is irreducible over $\mathbb{Z}_p$ then $\frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}$ is a field. If $f(x)$ is a reducible polynomial, then every non-zero element in $\frac{\mathbb{Z}_p[x] }{\langle f(x) \rangle}$ is either a zero-divisor or a unit. If we exclude the zero-divisors and zero, we have a finite Abelian group under multiplication denoted by $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big)$. Since every finite Abelian group is a direct product of cyclic groups of prime-power order, we can find the isomorphism class for $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big)$. We investigate the structure of $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big)$ for a prime $p$ and various $f(x)$. We conclude with some result on the structure of a certain family of subgroups of $U \Big( \frac{\mathbb{Z}_p[x]}{\langle f(x)\rangle} \Big)$

Groups of units of Zp[x] modulo f (x)

We give algorithms for expressing groups of units of polynomial rings over fields of prime order as direct products of subgroups

Représentations de l'espace en Occident de l'Antiquité tardive au XVIe s.

Gautier Dalché Patrick, Prontera Francesco, Westrem Martin Scott D. Représentations de l'espace en Occident de l'Antiquité tardive au XVIe s.. In: École pratique des hautes études. Section des sciences historiques et philologiques. Livret-Annuaire 17. 2001-2002. 2003. pp. 147-153

Représentations de l'espace en Occident de l'Antiquité tardive au XVIe s.

Gautier Dalché Patrick, Prontera Francesco, Westrem Martin Scott D. Représentations de l'espace en Occident de l'Antiquité tardive au XVIe s.. In: École pratique des hautes études. Section des sciences historiques et philologiques. Livret-Annuaire 17. 2001-2002. 2003. pp. 147-153