Self-Avoiding Walks and Fibonacci Numbers

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

The Bisection Method: Which Root?

No abstract provided in this article

An Amazing Mathematical Card Trick

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 if the pair should be flipped over or not. He goes through the pile again four cards at a time and before each group of four is dealt to a pile, the audience gets to decide whether each group should be flipped over or not. Then the cards are dealt into four rows of five cards. The audience can decide, for each row, whether it should be dealt from left to right or from right to left, producing an arrangement like the one shown

Book Review: Across the Board: The Mathematics of Chessboard Problems by John J. Watkins

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

Sensible Rules for Remembering Duals -- The S-O-B Method

We present a natural motivation and simple mnemonic for creating the dual LP of any linear programing problem

Mathematical Constance (A Poem Dedicated to Constance Reid)

Mathematical Constance (A Poem Dedicated to Constance Reid) I think that I shall never see A constant lovelier than e, Whose digits are too great too state, They\u27re 2.71828… And e has such amazing features It\u27s loved by all (but mostly teachers). With all of e\u27s great properties Most integrals are done with … ease. Theorems are proved by fools like me But only Euler could make an e. I suppose, though, if I had to try To choose another constant, I Might offer i or phi or pi. But none of those would satisfy. Of all the constants I know well, There\u27s only one that rings the Bell. Not pi, not i, nor even e. In fact, my Constance is a she. It\u27s Constance Reid, I would not fool ya\u27 With Books like Hilbert, Courant, and Julia. Of all the constants you will need, There’s only one that you should Reid

Combinatorics and Campus Security

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 pressed simultaneously and / or successively, but the same button cannot be used more than once in the same combination. I had a student (obviously not a math major) email me that there are only 120 possibilities, but even I know this is only if you press all five buttons one at a time. It doesn\u27t take into account 1-23-4-5, for instance. My question to you is how many combinations exist, and is it enough to keep our buildings adequately protected? To clarify, combinations like 1-25-4 (which is the same as 1-52-4 but different from 4-25-1) and 1-2-5-43 are legal, whereas 13-35 is illegal because the number 3 is used twice. I gave this problem to the students in my discrete mathematics class as a bonus exercise. Most arrived at the (correct) answer of 1081 or 1082 by breaking the problem into oodles of cases, but this would not have been a convenient method if the locks contained 10 buttons instead of 5. I use this problem as an excuse to demonstrate the power of generating functions by solving the n-button problem. Most students are amazed that the problem is essentially solved by the function ex /(2 - ex), which leads to a surprisingly accurate approximation of n!/(ln 2)n+1

Proof With Words: 2 + 11 - 1 = 12

Proof with words: 2 + 11 – 1 = 12 TWo ELeVE

Squaring, Cubing, and Cube Rooting

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\u27s chapter on Lightning Calculators. It was used by the mathematician A. C. Aitken to mentally square large numbers

Proof Without Words: Alternating Sums of Odd Numbers

Proof for alternating sums of odd numbers in two figures