We explore some magical properties of Fibonacci numbers, and a combinatorial interpretation of them which reveals most of their hidden secrets. It is easy to show that the number of ways to tile a 1-by-n board with squares and dominoes is the nth Fibonacci number f_n where f_1=1 and f_2=2. It is less well-known that the number of ways to tile a 1-by-n bracelet with squares and dominoes is the nth Lucas number. In fact, similar combinatorial interpretations can be given for all positive Fibonacci-like sequences as well as to the numerators and denominators of all finite simple continued fractions. Armed with these interpretations, and the occasional use of probability, many beautiful identities of number theory can be viewed as stories that are ``recounted'' in two different ways. (This talk is based on joint work with Jennifer Quinn and Francis Su.)
Arthur Benjamin is a "lightning calculator" and will demonstrate and explain his secrets of rapid mental calculation. He has presented this mixture of math and magic to audiences all over the world.
Arthur Benjamin earned his B.S. in Applied Mathematics from Carnegie Mellon U and his PhD in Mathematical Sciences from Johns Hopkins University. Since 1989, he has taught at Harvey Mudd College, where he is currently Professor of Mathematics. He has served as Editor of the Spectrum book series for MAA, and currently serves on the editorial board of Mathematics Magazine, the UMAP Journal, and the Anneli Lax New Mathematical Library. In 2000, he received the MAA's Haimo Award for Distinguished Teaching. Aside from his research interests in combinatorics and game theory, he enjoys tournament backgammon, racing calculators, and performing magic.
|101||From US Highway 101, take the De La Cruz Boulevard/Santa Clara exi t and follow the signs to El Camino and the main campus entrance.|
|280||From I-280, take I-880 north toward Oakland to the Alameda exit. Turn left onto The Alameda (which becomes El Camino Real) to the main campus entrance.|
|880||From I-880, take the Alameda exit, travel north (The Alameda becomes El Camino Real) to the main campus entrance.|