Other Mathematics Commons^™

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

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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 always had a passion for mental calculation, and the following formula appears in Gardner's chapter on "Lightning Calculators." It was used by the mathematician A. C. Aitken to mentally square large numbers.

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

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We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.

Book Review: Across The Board: The Mathematics Of Chessboard Problems By John J. Watkins, Arthur T. Benjamin

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I think I became a mathematician because I loved to play games as a child. I learned about probability and expectation by playing games like backgammon, bridge, and Risk. But I experienced the greater thrill of careful deductive reasoning through games like Mastermind and chess. In fact, for many years I took the game of chess quite seriously and played in many tournaments. But I gave up the game when I started college and turned my attention to more serious pursuits, like learning real mathematics.

Mathematical Magic, Arthur T. Benjamin

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In this paper, we present simple strategies for performing mathematical calculations that appear magical to most audiences. Specifically, we explain how to square large numbers, memorize pi to 100 places and determine the day of the week of any given date.

Analysis Of The N-Card Version Of The Game Le Her, Arthur T. Benjamin, Alan J. Goldman

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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.

Proof With Words: 2 + 11 - 1 = 12, Arthur T. Benjamin

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Proof with words: 2 + 11 – 1 = 12

TWo ELeVEn

Almost Periodic Factorization Of Certain Block Triangular Matrix Functions, Ilya M. Spitkovsky, Darryl H. Yong

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Let

where , and . For rational such matrices are periodic, and their Wiener-Hopf factorization with respect to the real line always exists and can be constructed explicitly. For irrational , a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible and commuting , was disposed of earlier-it was discovered that an almost periodic factorization of such matrices does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when is not invertible but the commute pairwise (). The complete description is …

The Best Way To Knock 'M Down, Arthur T. Benjamin, Matthew T. Fluet '99

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"Knock 'm Down" is a game of dice that is so easy to learn that it is being played in classrooms around the world. Although this game has been effective at developing students' intuition about probability [Fendel et al. 1997; Hunt 1998], we will show that lurking underneath this deceptively simple game are many surprising and highly unintuitive results.

Bounds On A Bug, Arthur T. Benjamin, Matthew T. Fluet '99

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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 both in place, the remaining body parts can be obtained in any order by rolling two 3s for the eyes, one 4 for the nose, two 5s for the …

Optimal Klappenspiel, Arthur T. Benjamin, Derek Stanford '93

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The game Klappenspiel ("flipping game") is a traditional German game of flipping tiles according to dice rolls. In this paper, we derive the optimal strategy for this game by using dynamic programming. We show that the probability of winning using the optimal strategy is 0.30%.