Arthur T. Benjamin

Harvey Mudd College · Mathematics

Suppose we place checkers in the lower left corner of a Go board and wish to move them to the upper right corner in as few moves as possible, where the pieces move as in the game of Chinese checkers. Auslander, Benjamin, and Wilkerson in 1993 generalized this game for integer lattices and defined a measure of speed for a starting configuration of p...

While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down permutations.

We describe a “handy” method, due to John Conway, for quickly finding all relatively small prime factors of 3-digit and 4-digit numbers.

In this article, we consider the question “Given a list of odds on a horse race, is it possible, by betting the right amount on each horse, to win money regardless of the outcome of the race?” Converting the given odds to probabilities and summing those probabilities yields an easily calculated parameter that indicates whether the answer to this qu...

We prove the r-Fibonacci identities of Howard and Cooper [4] using a combinatorial tiling approach.

We present several effective ways for a magician to create a 4-by-4 magic square where the total and some of the entries are prescribed by the audience.

I still recall my thrill and disappointment when I read Mathematical Carnival, by Martin Gardner. I was thrilled because, as my high school teacher had recommended, mathematics was presented in a playful way that I had never seen before. I was disappointed because it contained a formula that I thought I had "invented" a few years earlier. I have al...

We introduce the function a(r,n) which counts tilings of length n+r that utilize white tiles (whose lengths can vary between 1 and n) and r identical red squares. These tilings are called two-toned tilings. We provide combinatorial proofs of several identities satisfied by a(r,n) and its generalizations, including one that produces kth-order Fibona...

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.

While formulas for the sums of kth binomial coefficients can be shown inductively or algebraically, these proofs give little insight into the combinatorics involved. We prove formulas for the sums of 3rd and 4th binomial coefficients via purely combinatorial arguments.

We provide a combinatorial proof of a formula for the sum of evenly spaced binomial coefficients, ∑k≥0(n/rk). This identity, along with a generalization, are proved by counting weighted walks on a graph.

We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m,n⩾0, Tm(Tn(x))=Tmn(x) and Um−1(Tn(x))Un−1(x)=Umn−1(x).

We provide a combinatorial proof of the trigonometric identity cos(nθ)=Tn(cosθ), where Tn is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind.

A magician gives a member of the audience 20 cards to shuffle. After the cards are thoroughly mixed, the magician goes through the deck two cards at a time, sometimes putting the two cards face to face, sometimes back to back, and sometimes in the same direction. Before dealing each pair of cards into a pile, he asks random members of the audience...

Combinatorial proofs are appealing since they lead to intuitive understanding. Proofs based on other mathematical techniques may be convincing, but still leave the reader wondering why the result holds. A large collection of combinatorial proofs is presented in (1), including many proofs of Fibonacci identities based on counting tilings of a one-di...

Determinantsof matrices involving the Catalan sequence have ap- peared throughout the literature. In this paper, we focus on the evaluation of Hankel determinants featuring Catalan numbers by counting nonintersecting path systems in an associated Catalan digraph. We apply this approach in order to revisit and extend a result due to Cvetkovic-Rajkov...

We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-t...

Chebyshev polynomials have several elegant combinatorial interpretations. Specifically, the Chebyshev polynomials of the first kind are defined by T0(x) = 1, T1(x) = x, and Tn(x) = 2x Tn−1(x) − Tn−2(x). Chebyshev polynomials of the second kind Un(x) are defined the same way, except U1(x) = 2x. Tn and Un are shown to count tilings of length n strips...

In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting...

Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we Describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule...

We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences Gn that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well. [1] A. T. Benjamin and J. J. Quinn,...

A combinatorial argument is used to explain the integrality of Fi-bonomial coefficients and their generalizations. The numerator of the Fibono-mial coefficient counts tilings of staggered lengths, which can be decomposed into a sum of integers, such that each integer is a multiple of the denominator of the Fibonomial coefficient. By colorizing this...

We begin by solving a problem posed from the Monthly. Show n Σ r=0 (−1)r(n r)(2n - 2r n - 1) = 0. We prove this combinatorially by pairing up objects that have an odd value of r with objects that have an even value of r. The proof, and its generalizations, lead to many new and interesting identities.

In this article, we pursue the reverse strategy of using probability to derive an and develop an exponential generating function for an in Section 3. In Section 4, we present a method for finding an exact, non-recursive, formula for an.

In (2), Neville Robbins explores many properties of the Lucas triangle, an in- finite triangular array with properties similar to Pascal's triangle. In this paper, we provide a combinatorial explanation for the entries of this triangle. This inter- pretation results in extremely quick and intuitive proofs of most of the properties (proved mostly by...

If you choose two polynomials at random, what is the chance that they are relatively prime? We first answer this question when the polynomials are nth degree polynomials whose coefficients come from the set {0, 1} under mod 2 arithmetic. Here, for any n ≤ 1, the chance that they are relatively prime is 1/2, independent of n. In our combinatorial pr...

We provide combinatorial interpretations for determinants which are Fibonacci numbers of several recently introduced Hessenberg matrices. Our arguments make use of the basic definition of the determinant as a signed sum over the symmetric group.

We provide combinatorial derivations of solutions to intertwined second order linear recurrences (such as an = pbn-1 + qan-2, bn = ran-1 + sbn-2) by counting tilings of length n strips with squares and dominoes of various colors and shades. A similar approach can be applied to intertwined third order recurrences with coefficients equal to one. Here...

The starting point of this capsule is counting the number of rectangles in an m × n checkerboard (with a neat four-line derivation). From this, the formula for the sum of the first n cubes of positive integers and other results are derived..

By combinatorial arguments, we prove that the number of self- avoiding walks on the strip {0,1} ◊Z is 8Fn 4 when n is odd and is 8Fn n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0,1}◊Z, Z◊Z, the triangular lattice, and the cubic lattice.

We fully concur with Richard Askey's February 2004 “Delving Deeper” column. Discovering and proving identities containing Fibonacci numbers can be satisfying for students and teachers alike. His article touched on multiple strategies including induction, linear algebra, and a hefty dose of algebraic manipulation to derive many interesting identitie...

These simple math secrets and tricks will forever change how you look at the world of numbers. Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends—and yourself—with incredible calculations you never thought you could master, as renowned "mathemagician" Arthur Benjamin shares his techniques fo...

In this paper, we provide combinatorial interpretations for some deter-minantal identities involving Fibonacci numbers. We use the method due to Lindström-Gessel-Viennot in which we count nonintersecting n-routes in carefully chosen digraphs in order to gain insight into the nature of some well-known determinantal identities while allowing room to...

Proof for alternating sums of odd numbers in two figures.

Let D(n) denote the number of derangements of n elements, i.e., the number of permutations of n elements without any fixed points. The formula for D(n) is well known, but how many of these derangements are even permutations? Let E(n) and O(n) denote the number of derangements of n elements that are even and odd permutations, respectively. Thus E(n)...

1. THE PROBLEM OF THE DETERMINED ANTS. Imagine four determined ants who simultaneously walk along the edges of the picnic table graph of Figure 1. The ants can move only to the right (northeast, southeast, and sometimes due east) with the goal of reaching four different morsels.

The Lucas numbers, 2, 1, 3, 4, 7, 11, 18, 29, 47,..., named in honor of Edouard Lucas (1842-1891), are defined by L0 = 2, L1 = 1, and Ln = Ln-1+ Ln-2for n = 2.It is easy to show that, for n = 1, Ln counts the ways to create a bracelet of length n using beads of length one or two, where bracelets that di er by a rotation or a reflection are still co...

In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued fractions. It uses an idea by Henry J. S. Smith ([3], [5], and [6]) most recently described in [4] (which provides a new proof of uniqueness and reprint...

This issue focuses on proving several interesting facts about the Fibonacci Sequence using a combinatorial proof. The aim of Delving Deeper is for teachers to pose and solve novel math problems, expand on mathematical connections, or offer new insights into familiar math concepts. Delving Deeper focuses on mathematics content appealing to secondary...

We present a combinatorial proof that the wheel graph W n has L 2n − 2 spanning trees, where L n is the nth Lucas number, and that the number of spanning trees of a related graph is a Fibonacci number. Our proofs avoid the use of induction, determinants, or the matrix tree theorem.

Using elementary properties of the Gaussian integers, all solutions to the Diophantine equation a 2 +b 2 =c n are succinctly described for all integer n≥2.

In [4], Carlitz demonstrates $$ {F_L}\sum\limits_{{x_1} = 0}^n {\sum\limits_{{x_2} = 0}^n {...\sum\limits_{{x_{L = 0}}}^n {\left( \begin{gathered}n - {x_L} \hfill \\{x_1} \hfill \\\end{gathered} \right)} } } \left( \begin{gathered}n - {x_1} \hfill \\{x_2} \hfill \\\end{gathered} \right)...\left( \begin{gathered}n - {x_L} - 1 \hfill \\{x_L} \hfill \...

Problem A — 6 from the 1990 Putnam exam states: If X is a finite set, let |X| denote the number of elements in X. Call an ordered pair (S,T) of subsets of {1, 2,... , n} admissible if s > |T| for each s ∈ S, and t > |S| for each t ∈ T. How many admissible ordered pairs of subsets of {1, 2,..., 10} are there? Prove your answer.

We present a tiling interpretation for k-th order linear recurrences, which yields new combinatorial proofs for recurrence identities. Moreover, viewing the tiling process as a Markov chain also yields closed form Binet-like expressions for these recurrences.

Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. in Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. T...

Hyperharmonic numbers arise by taking repeated partial sums of harmonic numbers. These numbers can be expressed in terms of r-Stirling numbers, leading to combinatorial interpretations of many interesting identities.

Presentation of two simple combinatorial proofs.

We present a complete solution to a card game with historical origins. Our analysis exploits the convexity properties in the payoff matrix, allowing this discrete game to be resolved by continuous methods.

In the game Knock 'm Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of mov...

We utilize the KOH theorem to prove the unimodality of integer partitions with at most a parts, all parts less than or equal to b, that are required to contain either repeated or consecutive parts. We connect this result to an open question in quantum physics relating the number of distinct total angular momentum multiplets of a system of N fermion...

. We present a tiling interpretation for k th order linear recurrences, which yields new combinatorial proofs for recurrence identities. Moreover, viewing the tiling process as a Markov chain also yields closed form Binet-like expressions for these recurrences. 1.

A Fermion to Boson transformation is accomplished by attaching to each Fermion a tube carrying a single quantum of flux oriented opposite to the applied magnetic field. When the mean field approximation is made in Haldane's spherical geometry, the Fermion angular momentum l_F is replaced by l_B=l_F-(N-1)/2. The set of allowed total angular momentum...

A Fermion to Boson transformation is accomplished by attaching to each Fermion a single flux quantum oriented opposite to the applied magnetic field. When the mean field approximation is made in the Haldane spherical geometry, the Fermion angular momentum $l_F$ is replaced by $l_B= l_F-{1\over2}(N-1)$. The set of allowed total angular momentum mult...

Using path counting arguments, we prove This inequality, motivated by graph coloring considerations, has an interesting geometric interpretation.

In this paper we introduce t.he concept of colored Fi- bonacci tilings which leads to charming ~ombinat~orial prook of Fi- bonacci and Lucas number identites. Cornbinatorial proofs can lead to a greater appreciatioli arid understand- ing for any topic. Fibonacci arid Lucas identities are no exception. Let F7,, = F7,,-1 + Fn,-2, where Fo = 0, F1 = 1...

this paper, we provide a combinatorial interpretation for the numerators and denominators of continued fractions which makes this reversal phenomenon easy to see. Our interpretation also allows us to visualize many important identities involving continued fractions. We begin by defining some basic terminology. Given an infinite sequence of integers...

In the game of Cootie, players race to construct a "cootie bug" by rolling a die to collect component parts. Each cootie bug is composed of a body, a head, two eyes, one nose, two antennae, and six legs. Players must first acquire the body of the bug by rolling a 1. Next, they must roll a 2 to add the head to the body. Once the body and head are bo...

To get around an old California law that prohibits the game of "21," California card casinos introduced a variation of standard Blackjack, called LA Blackjack, in which the objective is to get as close as possible to 22 without going over. The standard game of Blackjack, or "21", pits the player against the dealer ("the house"). Money lost by the p...

We interpret generalized Fibonacci numbers as phased tilings and introduce several combinatorial techniques which provide new proofs for a host of identities. These follow naturally as the phased tilings are counted, represented, and transformed in clever ways.

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph Sm(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained...

Like many games, people place money wagers on backgammon games. These wagers can change during the game. In order to make intelligent bets, one needs to know the chances of winning at any point in the game. We were working on this for positions near the end of the game when we needed to explicitly label each of the positions so the computer could r...

One day I received electronic mail from our director of campus security [Gilbraith 1993]: "I have a puzzle for you that has practical applications for me. I need to know how many different combinations there are for our combination locks. A lock has 5 buttons. In setting the combination you can use only 1button or as many as 5. Buttons may be press...

A stable marriage problem of size 2n is constructed which contains stable matchings. This construction provides a new lower bound on the maximum number of stable matchings for problems of even size and is comparable to a known lower bound when the size is a power of 2. The method of construction makes use of special properties of the latin marriage...