## Thursday, March 31, 2016

### Abstracts to the papers that appeared in Journal of Recreational Mathematics 36(3)

Abstracts to the papers that appeared in Journal of Recreational Mathematics 36(3)

The Hypofactorial
N. E. Myridis
Aristotle University of Thessaloniki, Greece

Abstract
We introduce in this article a new arithmetic operation and call it hypofactorial. This operation is the counterpart of the known factorial operation. The hypofactorial reveals some important and useful properties and relations. Among them we note for instance that the hypofactorial enables the evaluation of factorials of non-integral numbers.

A Binomial Identity with Catalan Dividends
Thomas Koshy and Angelo DiDomenico
Framingham University

Abstract
We present an integer function using binomial coefficients and employ it to develop an identity involving them. This identity yields several interesting properties of the well known Catalan numbers, where n ≥ 0.

C(n) = [1/(n + 1)] * Combination(2n,n).

SumSum Puzzles
Frank Rubin

Abstract
SumSum is a new numerical logic puzzle which combines elements of Sudoku, Kakuro and diagramless crossword puzzles.  Like Kakuro, you need to fill the digits 1 to N into the white spaces to achieve the specified row and column sums.  Like Sudoku, each digit must appear exactly once in each row and each column.  The twist is that the black spaces are not shown. You need to figure out which spaces are white and which are black.

Twenty-One Forever!
Colin Foster
King Henry VIII School, UK

Abstract
This article explores mathematical ways of legitimately claiming to be 21 years old for every age beyond 21. Modular arithmetic is first considered, so, for instance, a 48-year-old could say that they are “21 mod 27”. Secondly, different number bases are examined, so, for instance, a 37-year-old could say that they are "21 in base 18”. These ideas are extended to the multiplication tables.

What? Four!
Steven Kahan
Queens College

Abstract
In this paper a numerical identity that concludes with a 4 is developed from a basic triangle using the Law of Cosines and Heron’s Formula.

On a Probability Problem of Charles Dodgson
Robert J. MacG. Dawson
Saint Mary’s University, Nova Scotia Canada
rdawson@cs.stmarys.ca

Abstract
Charles Dodgson composed a large number of mathematical problems, many
of which were published during his lifetime.  One which was found among
the papers of one of his friends (without solution) and published
posthumously involves determining the probability of a certain event
given the joint testimony of a number of disagreeing witnesses, supposed
to be "equally reliable." In his book "Recovered Lewis Carroll Puzzles",
Wakeling gives a paradoxical (and in fact incorrect) solution. We give a
correct solution, show that although the paradox does not appear it
could have for other parameter values, and argue on this evidence that
Dodgson (whose grasp of continuous probability theory was, on the
evidence of his published problems, weak)  probably did not know the

Two Conjectures on Prime Numbers
Shantanu Dey and Moloy De

Abstract
Prime numbers and their distribution have been fascinating mathematicians, both professionals as well as amateurs for ages. In this short note we describe two conjectures on prime numbers. We think these two conjectures will throw new light on the distribution of prime numbers.

Railway Mazes: From Picture to Solution
Antonin Slavik
Charles University, Czech Republic
slavik@karlin.mff.cuni.cz
Stan Wagon
Macalester College

Abstract
Railway mazes are convoluted mazes based on how a train travels along tracks; they were invented by Lionel Penrose. We describe an algorithm to solve such mazes, and show how the algorithm can be made to work with input that is simply a photograph of the maze. We apply this method to solve the more complicated railway mazes that were developed by L. Penrose and his son, Sir Roger Penrose.

Which Winning Streak in College Basketball Would UConn-Sider the Best?
Alexandra M. Rotatori, Kathryn K. Logan, Alison J. McAnaney and Paul M. Sommers
Middlebury College

Abstract
The three longest winning streaks in college basketball belong to the University of California,          Los Angeles (UCLA) Bruins men’s team (88 consecutive wins) and the University of Connecticut (UConn) Huskies women’s team (70 and 90).  The authors examine the average margin of victory as a percentage of the total number of points scored by both teams in each game of the three longest winning streaks.   This average margin of victory metric is calculated for home games, away games, conference and non-conference games.  The 70- and 90-game consecutive winning streaks set by the two Connecticut women’s teams reveal significantly higher average margins than that for the men’s UCLA streak.

Is A Defensive Tackle Worth More Than A Quarterback?
Connor E. Green and Paul M. Sommers
Middlebury College

Abstract
Approximate Value (AV) assesses player worth at any position in the National Football League (NFL).  The weighted career AV (WCAV) can be used to compare career productivity of players at different positions.   The authors examine the WCAV from all seven rounds of five draft classes, between 1986 and 1990 (for which no drafted player is still active in the NFL), and compare the average WCAV of one position to that of another.  A regression analysis that controls for differences in career length suggests that defensive tackles, offensive tackles, and linebackers are most valuable; quarterbacks and tight ends (after allowing for differences in career length) are least valuable.

Have Bonus Points Affected Scoring in the Rugby World Cup?
Mwaki Harri Magotswi and Paul M. Sommers
Middlebury College

Abstract
Beginning in 2003, the Rugby World Cup awarded a bonus point during the first round of play (the pool or group stage) if (i) a team scored four or more tries in a match or (ii) the losing team lost the match by seven or fewer points.  The authors assess the impact of the bonus point system on scoring in pool play in two World Cups before and after implementation of the bonus point system.  The before-and-after comparisons show no discernible effect on the average number of tries or points scored per match or on the relative frequency of tries (as a percentage of all scoring plays) per match.

What’s the Spin on Serve Speed and Different Court Surfaces?
David A. Farah, Johann N. Riefkohl, Alexandra L. McAtee and Paul M. Sommers
Middlebury College

Abstract
The authors examine the paired difference between the winner’s and loser’s average first and second serve speeds in each match of each round of play for women’s and men’s singles at the 2010 U.S. Open (hard court surface) and the 2011 French Open (slower clay court).  For the U.S. Open, women who won their matches had (through the first four rounds) significantly faster average first and second serve speeds than their opponents.  For the French Open, men who won their matches had (through only the first two rounds) significantly faster first serve speeds.  Average serve speed was not greater for match winners than for losers on either court surface beyond the fourth round.