Harold Reiter | University of North Carolina at Charlotte (original) (raw)

Papers by Harold Reiter

Discrete Mathematics & Theoretical Computer Science, 2005

The purpose of this paper is to solve a special class of combinational games consisting of two-pi... more The purpose of this paper is to solve a special class of combinational games consisting of two-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the games. Two players alternate moving. Each player in his turn first chooses one of the piles, and his choice of piles can change from move to move. He then removes counters from this chosen pile. A function f : Z + → Z + is given which determines the maximum size of the next move in terms of the current move size. The game ends as soon as one of the two piles is empty, and the winner is the last player to move in the game. The games for which f (k) = k, f (k) = 2k, and f (k) = 3k use the same formula for computing the smallest winning move size. Here we find all the functions f for which this formula works, and we also give the winning strategy for each function. See [7] for a discussion of the single pile game.

After first defining weighted centroids that use complex ari thmetic, we then make a simple obser... more After first defining weighted centroids that use complex ari thmetic, we then make a simple observation which proves Theorem 1. We n ext define complex homothety. We then show how to apply this theory to tr iangles (or polygons) to create endless numbers of homothetic triangle s (or polygon). The first part of the paper is fairly standard. However, in the fina l part of the paper, we give two examples which illustrate that examples can easily be given in which the simple basic underpinning is so disguised that it is not a t all obvious. Also, the entire paper is greatly enhanced by the use of complex ari thmetic.

The purpose of this paper is to solve a class of combinatorial games consisting of one-pile count... more The purpose of this paper is to solve a class of combinatorial games consisting of one-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the game. The maximum size of the move is determined by a move function f whose arguments are pile sizes. In another paper[8], we will discuss the game in which the number of counters that can be removed depends on the number removed in the previous move.

Missouri Journal of Mathematical Sciences, 2012

Let P (x) and Q (x) be polynomials of degrees n and m respectively. Let P i (x) and Q j (x) denot... more Let P (x) and Q (x) be polynomials of degrees n and m respectively. Let P i (x) and Q j (x) denote the ith, jth derivatives of P (x) and Q (x). Define the (m + n) × (m + n) matrix M (x) as follows.

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

Mathematics Teacher, 2012

This method for counting lattice octagons strengthens students' counting skills and geometric... more This method for counting lattice octagons strengthens students' counting skills and geometrical thinking.

Mathematics Magazine, 1996

Rocky Mountain Journal of Mathematics, 1972

Let (X, τ) be a Hausdorff space, where X is an infinite set. The compact complement topology τ on... more Let (X, τ) be a Hausdorff space, where X is an infinite set. The compact complement topology τ on X is defined by: τ = {∅} ∪ {X \ M : M is compact in (X, τ)}. In this paper, properties of the space (X, τ) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.

Abstract. Basically there are two types of games, namely games that do and games that do not invo... more Abstract. Basically there are two types of games, namely games that do and games that do not involve chance. Classical n-pile Bouton's nim is an example of a game that does not involve chance. The coin matching game is a game that does involve chance. In the coin ...

Mathematics and Computer Education, 1991

Abstract: Annotated are questionnaire responses from 88 state and regional contest supervisors co... more Abstract: Annotated are questionnaire responses from 88 state and regional contest supervisors concerning trends in calculator usage and nonusage among 59 represented mathematical contests. Discussed are advantages and disadvantages for specific contest ...

The College Mathematics Journal, 1991

Page 1. STUDENT RESEARCH PROJECTS EDITOR: Irl C. Bivens Department of Mathematics Davidson Colleg... more Page 1. STUDENT RESEARCH PROJECTS EDITOR: Irl C. Bivens Department of Mathematics Davidson College Davidson, NC 28036 A student research project is an open-ended question or set of questions that is intended ...

Citeseer

Introduction. In this paper, two players alternate removing a positive number of counters from on... more Introduction. In this paper, two players alternate removing a positive number of counters from one of n piles of counters, and the choice of which pile he removes from can change on each move. On his initial move, the player moving first can remove from one pile of his choice ...

Citeseer

A standard technique for solving the recursion xn+1 = g (xn) where g : C → C is a complex functio... more A standard technique for solving the recursion xn+1 = g (xn) where g : C → C is a complex function is to first find a fairly simple function g : C → C and a bijection f : C → C such that g = f◦g◦f−1 where ◦ is the composition of functions. Then xn = gn (x0)=(f ◦ gn ◦ f−1)(x0) where gn and gn are ...

Discrete Mathematics & Theoretical Computer Science, 2005

The purpose of this paper is to solve a special class of combinational games consisting of two-pi... more The purpose of this paper is to solve a special class of combinational games consisting of two-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the games. Two players alternate moving. Each player in his turn first chooses one of the piles, and his choice of piles can change from move to move. He then removes counters from this chosen pile. A function f : Z + → Z + is given which determines the maximum size of the next move in terms of the current move size. The game ends as soon as one of the two piles is empty, and the winner is the last player to move in the game. The games for which f (k) = k, f (k) = 2k, and f (k) = 3k use the same formula for computing the smallest winning move size. Here we find all the functions f for which this formula works, and we also give the winning strategy for each function. See [7] for a discussion of the single pile game.

After first defining weighted centroids that use complex ari thmetic, we then make a simple obser... more After first defining weighted centroids that use complex ari thmetic, we then make a simple observation which proves Theorem 1. We n ext define complex homothety. We then show how to apply this theory to tr iangles (or polygons) to create endless numbers of homothetic triangle s (or polygon). The first part of the paper is fairly standard. However, in the fina l part of the paper, we give two examples which illustrate that examples can easily be given in which the simple basic underpinning is so disguised that it is not a t all obvious. Also, the entire paper is greatly enhanced by the use of complex ari thmetic.

The purpose of this paper is to solve a class of combinatorial games consisting of one-pile count... more The purpose of this paper is to solve a class of combinatorial games consisting of one-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the game. The maximum size of the move is determined by a move function f whose arguments are pile sizes. In another paper[8], we will discuss the game in which the number of counters that can be removed depends on the number removed in the previous move.

Missouri Journal of Mathematical Sciences, 2012

Let P (x) and Q (x) be polynomials of degrees n and m respectively. Let P i (x) and Q j (x) denot... more Let P (x) and Q (x) be polynomials of degrees n and m respectively. Let P i (x) and Q j (x) denote the ith, jth derivatives of P (x) and Q (x). Define the (m + n) × (m + n) matrix M (x) as follows.

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

American Mathematics Competitions 1995–2000 Contests

Mathematics Teacher, 2012

This method for counting lattice octagons strengthens students' counting skills and geometric... more This method for counting lattice octagons strengthens students' counting skills and geometrical thinking.

Mathematics Magazine, 1996

Rocky Mountain Journal of Mathematics, 1972

Let (X, τ) be a Hausdorff space, where X is an infinite set. The compact complement topology τ on... more Let (X, τ) be a Hausdorff space, where X is an infinite set. The compact complement topology τ on X is defined by: τ = {∅} ∪ {X \ M : M is compact in (X, τ)}. In this paper, properties of the space (X, τ) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.

Abstract. Basically there are two types of games, namely games that do and games that do not invo... more Abstract. Basically there are two types of games, namely games that do and games that do not involve chance. Classical n-pile Bouton's nim is an example of a game that does not involve chance. The coin matching game is a game that does involve chance. In the coin ...

Mathematics and Computer Education, 1991

Abstract: Annotated are questionnaire responses from 88 state and regional contest supervisors co... more Abstract: Annotated are questionnaire responses from 88 state and regional contest supervisors concerning trends in calculator usage and nonusage among 59 represented mathematical contests. Discussed are advantages and disadvantages for specific contest ...

The College Mathematics Journal, 1991

Page 1. STUDENT RESEARCH PROJECTS EDITOR: Irl C. Bivens Department of Mathematics Davidson Colleg... more Page 1. STUDENT RESEARCH PROJECTS EDITOR: Irl C. Bivens Department of Mathematics Davidson College Davidson, NC 28036 A student research project is an open-ended question or set of questions that is intended ...

Citeseer

Introduction. In this paper, two players alternate removing a positive number of counters from on... more Introduction. In this paper, two players alternate removing a positive number of counters from one of n piles of counters, and the choice of which pile he removes from can change on each move. On his initial move, the player moving first can remove from one pile of his choice ...

Citeseer

A standard technique for solving the recursion xn+1 = g (xn) where g : C → C is a complex functio... more A standard technique for solving the recursion xn+1 = g (xn) where g : C → C is a complex function is to first find a fairly simple function g : C → C and a bijection f : C → C such that g = f◦g◦f−1 where ◦ is the composition of functions. Then xn = gn (x0)=(f ◦ gn ◦ f−1)(x0) where gn and gn are ...