Japheth Wood

Author: japheth

  • Math Physics Explore

    Excerpt from website: http://www.mathphysicsexplore.org

    We develop a curiosity and interest in mathematics so students, parents and teachers can improve their math and science education. Activities focus on various topics in mathematics and their application to physical phenomenon.

    This website and our exploratorium:

    • Encourage each person to reach his/her potential
    • Provide guidance for children participating in science fairs
    • Help teachers to explain math concepts
    • Provides travel exhibits for schools

    We are located at 1054 Freedom Plains Road, Poughkeepsie, NY 12603.

  • Bard Math Circle

    The Bard Math Circle primarily targets middle school and elementary school students in the Mid-Hudson Valley region of New York State.
    In Kingston, NY (a small city in which 15% of residents live below the poverty line and 27% of children live below the poverty line) , the target audience for the past two years has been members of the Boys & Girls Club, whose members are primarily minority youth from low socio-economic households. The highlight of the activities in Kingston was a group visit to the Bard campus, in which the Boys & Girls Club members participated in math workshops led by Bard College math professors. In 2010, the primary location is the Kingston Library in Midtown Kingston and targets a wider middle school audience, a diverse socio-economic, racial and gender balanced group.
    In Tivoli, (a semi-rural community north of Bard College) the Bard Math Circle holds monthly sessions at the Tivoli Library, The audience consists of families with children in middle school and younger.
    In Red Hook, (a suburban and semi-rural rural community east and south of Bard College), we also target a middle school audience, in partnership with a math teacher at the Linden Avenue Middle School.

    Origins

    The Bard Math Circle was formed in 2007 by Bard College Professors Lauren Rose and Japheth Wood and then students Shelley Stahl and Ezra Winston. The Bard Math Circle is run jointly by students, under Bard’s Trustee Leader Scholar Program, and faculty at Bard College.

  • Limaçon 2011

    Long Island Math Conference
    Nassau County Math Teachers Association

    Date: Friday, March 18, 2011
    Time: 7:45 A.M. to 2:35 P.M.
    Location: SUNY College at Old Westbury Campus Center
    URL: http://ncmta.net/limacon.htm

    This is the annual Long Island Math Teachers’ Conference.

  • Rubik’s Cube Conversation over Breakfast

    If you read my previous post then you know that I’m at HCSSiM for the second half of the second half, teaching a mini on the Fundamental Theorem of Algebra. I’m very impressed with everything, from the teaching faculty to the students, and am really enjoying soaking in interesting mathematics every day. There is really high level mathematics going on here, and the students are truly immersed in mathematical knowledge and culture.

    Conversation at breakfast with Lucas and Gabe.

    I walked into a conversation about Rubik’s Cube records this morning at breakfast. Talking about new cube records, Lucas complained about kids who ask him if he solves the cube by just doing the same sequence of moves over and over again. Of course not! This would only work in a cyclic group!

    However, I argued that it is possible, if you look at it another way. Suppose that g is a long sequence of moves that traverses through all possible cube positions. Then you only have to do the sequence g once, and somewhere along the way, you’ll have solved the cube. Notice that the end result of performing the moves in sequence g is the identity permutation on the cube.

    We can improve this by finding a cube permutation g that generates a large cyclic subgroup of the cube group. Let G be the cyclic subgroup generated by g. If we can express g as a long sequence of cube moves that traverses through a complete set of coset representatives of G, then we have the cube neophyte’s dream: a sequence of cube moves, that it you do over and over again, will eventually solve the cube (of course, in the worst case scenario you’ll move through all possible configurations of the cube, but I’m not making any claim about the efficiency of this method!)

    We finished our breakfast conversation by posing a more reasonable problem: try this for a small group.

    1. Show that g = (1, 2, 3)(4, 5) is an element of S_5 of maximal order.
    2. Find a sequence of 120/6 = 20 permutations s_1, s_2, …, s_20 whose product is g, and whose partial products (s_1), (s_1 s_2), (s_1 s_2 s_3), …, (s_1 s_2 s_3 … s_20) is a set of coset representatives of the cyclic subgroup .
    3. Solve problem 2, where each s_i is a transposition.
    4. Solve problem 2, where each s_i is a transposition of adjacent elements.

    This seems like a good start to investigating the breakfast conversation problem. Let me know what you think!

  • NYMC Summer Workshop 2010 at Bard College

    Dates: July 26-30, 2010
    Location: Bard College, Annandale-on-Hudson, NY
    Theme: The Theorem of Pythagoras

    This week-long residential program is focused on creative and insightful mathematical problem solving for middle school math teachers. The high level mathematics activities and classes are led by NYMC instructors and Bard math professors.
    Participants will engage in a wide variety of mathematical investigation, problem solving, and classes that explore and develop a deep appreciation and understanding of the Theorem of Pythagoras and its consequences. Specific topics include geometric dissections, quadratic number system extensions, Diophantine equations, historical development, and a study of Euclid’s original proof, which dates back over 2000 years.

    Webpage: http://nymathcircle.org/2010workshop

  • HCSSiM 2010: Alive and Well!



    I visited the Hampshire College Summer Studies in Mathematics (    HCSSIM) program for the annual YP Day, and am happy to report the program is as vibrant as ever this summer.

    Yellow Pigs Day

    For those not familiar with HCSSiM lore and what YP day is, the initials YP stand for Yellow Pig, one of the two mascots of the program. David Kelly, the director of the program, has perhaps the largest collection of yellow pigs in the world.

    The other mascot of the program is 17, the favorite number of scads of summer studies students and alumni. YP Day (Yellow Pig’s Day) falls each year on July 17th, and is the traditional time for alumni to revisit the campus and reconnect with HCSSiM.

    This Year’s Faculty

    Arriving at HCSSiM on Friday in time for lunch, I met the faculty. In addition to Kelly, we have Rob Hochberg and Josh Greene also leading the workshops. Rob, a computer scientist, and I go way back to the 1980’s when we were both summer studies students, but this was my first time meeting Josh, a Columbia University mathematician, who attended HCSSiM in the 1990’s. To round out the crew we have the amazing:
    • Gabe, recent University of Chicago graduate
    • Misha, math graduate student at CUNY
    • Nadine, math graduate student at the University of Pittsburgh
    • Achyut, Hampshire College physics major
    • Peter, Harvey Mudd math major
    • Nate, U Mass math major
    • Julia, recent Hampshire College graduate
    • Emil, Harvey Mudd math major
    and of course the indefatigable Susan Goff, who has been the program assistant for over a decade.

    Prime Time Theorem

    I got my chance to meet the students later that afternoon, when I explained How to Solve a Cubic to them as Friday’s Prime Time Theorem speaker. The students are as sharp as ever, and an enthusiastic audience. I ended my talk with some open-ended questions.
    • Is there a quartic formula?
    • Is there a quintic formula?
    • Does every polynomial have a zero?
    Although most of the summer studies students know the general answer, I hope that these questions and my talk will pique their interest for my Mini later this summer on the Fundamental Theorem of Algebra. Apparently another Mini will focus on Galois Theory, so HCSSiM definitely has this topic covered!

    After dinner I had a chance to talk with the students during their problem session. They’re enjoying some Algebraic Number Theory, Cardinality Proofs, and Group Theory.

    Saturday, July 17th

    Yellow Pig’s Day fell on a Saturday this year, just perfect timing for the traditional Ultimate Frisbee Game, Kelly’s talk on The Mathematical and Social History of 17, the Yellow Pig Cake and the Singing of Traditional Yellow Pigs Day Carols.

    Saturdays start off with the morning workshops, though. I was able to wander between the three workshops and observe a wonderful proof of Wilson’s Theorem, a listing of Group Theory axioms (after two weeks of examples), and some interesting Cardinality Proofs.

    Mathcamp

    Surprisingly close this summer is Mathcamp, another wonderfully rich math environment for high school students. Math camp is just a 15 minute drive from Hampshire College at Mount Holyoke College, I visited Dan Zaharopol’s class on Real Analysis, and Alison Miller’s class on p-adic numbers.

    Among the more than 100 Mathcamp students, there are several who previously studied at HCSSiM, and I was intrigued discussing their positive experiences at both programs.

    Ultimate Frisbee Game, HCSSiM 2010 vs. HCSSiM <>

    My visit to HCSSiM 2010 ended with a grueling game of ultimate in excruciating heat. The alums were outnumbered three-to-one, but did well until the 2010 students developed a strategy that took advantage of their strength in numbers: they lined up and passed the frisbee down the field. Luckily, the aging alums still managed to come out ahead by dividing each team’s score by the number of players, and then multiplying by the average age.I drove back home soon after dinner, and enjoyed a beautiful double rainbow all the way through the Berkshires. I’m looking forward to returning in a few weeks to lead a Mini on The Fundamental Theorem of Algebra.

  • MAA Prep — Algebraic Number Theory

    June 28-July 2, 2010, Williams College, Williamstown, MA
    This week-long workshop is run by math professors John Cullinan (Bard) and Allison Pacelli (Williams), with a focus on Unique Factorization and its extensions, especially in quadratic number fields.
    http://math.bard.edu/cullinan/prep/

  • ¡Juguemos Kenken!

    (This column originally appeared in the July 2009 issue of La Voz, in Spanish.)

    Welcome back to the puzzle column. This time we have two new kenken puzzles for you. The first one is a 4×4 kenken that uses only addition, and the second a 5×5 kenken made up entirely of multiplication.

    For your convenience, we repeat the rules of kenken:

    Similar to Sudoku, your task is to fill in the 4×4 grid using only the numbers 1, 2, 3 and 4 (and the 5×5 grid using only the numbers 1,2,3,5 and 5) in such a way that:
    1. Each number appears only once per row.
    2. Each number appears only once per column.
    3. The numbers in each cage, when combined with the operation given, yield the target number.

    Since kenken means “wisdom squared” in Japanese, we know you’ll get a little smarter by successfully completing these puzzles!

    Multiplying

    The second kenken puzzle is entirely multiplication, which means that a good strategy to use is factorization. Look at the top middle cage, for example. We have to find three numbers whose product is 20. Multiplication has the amazing property that the order of the factors doesn’t matter, so listing the factors in non-decreasing order, there are four ways: 20 = 1×1×20 = 1×2×10 = 1×4×5 = 2×2×5. Of course we immediately eliminate the first two ways, since we are only allowed to fill in the grid with the numbers 1, 2, 3, 4 and 5.

    Using factorization on the bottom right cage is even more effective: 25 = 1×1×25 = 1×5×5. The first factorization uses 25, which is too big. The second factorization has a repeated factor, which in combination with rules 1 and 2 above, allow us to conclude the unique way to fill in the cage.

    Factorization leads to some amazing patterns. Consider the cage in the middle of the puzzle that prompts us to express 12 as the product of two numbers. The solutions are 12 = 1×12 = 2×6 = 3×4. The numbers that appear, 1, 2, 3, 4, 6 and 12, are called the divisors of 12. There is an amazing way to arrange the divisors of 12, shown below. Moreover, by counting 1 and the number 12 itself, 12 has 6 divisors.

    This figure is called the divisor lattice of 12. Notice that moving up and to the right (northeast) is equivalent to multiplying by 3. Likewise, moving up and to the left is equivalent to multiplying by 2. From a tilted perspective, the divisors of 12 form a 2×3 grid.

    Problem C: Draw the divisor lattices for the numbers from 1 to 20. What do 2, 3, 5 and 7 have in common? Can you draw the divisor lattice for 30? Is 1 a prime number? What other patterns do you notice?

    Problem D: What is the smallest number with exactly 2009 divisors? (count 1 and the number as divisors).

    If you enjoyed these puzzles, I’d love to hear from you. Please send in your solutions to any of the problems A, B, C or D as a comment.

  • ¡Juguemos Kenken!

    (This column originally appeared in the June 2009 issue of La Voz)

    KenKen es un puzzle nuevo inventado por Tetsuya Miyamoto, un maestro de matemáticas del Japón. KenKen significa “sabiduría al cuadrado” y se lo presenta como una variante del Sudoku. Este mes les traigo dos juegos de KenKen de 4×4. Uno más fácil con sumas solamente, y otro con más operaciones. ¡Y unas preguntas extra para los amantes de los rompecabezas matemáticos!

    ¿Qué es un juego KenKen?
    Cada juego KenKen de 4×4 consiste en una grilla cuadrada de 4×4 dividida en grupos llamados jaulas o regiones. Las regiones vienen en diferentes tamaños: 1, 2, 3 y a veces 4 cuadrados. Cada región indica el número al que se debe llegar y una operación.
    Por ejemplo, acá hay una región hecha con dos cuadrados, con un “7+” chiquito escrito en un rincón. Esto quiere decir que el número al que se debe llega es el 7 y la operación es la suma.

    La tarea es llenar la grilla de 4×4 usando solamente los números 1, 2, 3 y 4, de tal manera que:
    Cada número aparezca sólo una vez por fila.
    Cada número aparezca sólo una vez por columna.
    Los números en cada región, al combinarse con la operación dada, den por resultado el número al que se debe llegar.
    Volviendo a nuestro ejemplo, vemos que hay exactamente dos maneras posibles de completar la región: 7 = 3 + 4 y 7 = 4 + 3 son las únicas posibilidades. La elección correcta dependerá de las primeras dos reglas, y las posibilidades para las otras regiones.

    or
    A veces una región tiene un solo cuadrado. En ese caso, el número final es dado sin ninguna operación. Esto es un regalo gratis, y es una buena manera de comenzar a completar el rompecabezas.
    Esto es todo lo que necesitas para resolverlo, además de imprimir esta página para poder escribir, claro. ¡Así que a jugar KenKen!
    Problema A

    Problema B

    El cuadrado latino
    Un cuadrado de 4×4 completado con los números del 1 al 4 se llama cuadrado latino si satisface las dos condiciones de arriba. O sea, un cuadrado es latino si usa cada número sólo una vez en cada columna y en cada fila.
    Problema C: ¿Cuántos diferentes cuadrados latinos puede haber si la primer fila es 4 2 1 3?
    Puedes construir tu propio juego de KenKen empezando por el final. Primero, completa una grilla de 4×4 con un cuadrado latino. Segundo, dibuja regiones dentro de la grilla y elije una operación para cada región. Puedes empezar con la suma, para hacerlo fácil, pero los juegos de KenKen también tienen resta, multiplicación y división.
    En cada región, combina los números con la operación que elegiste para determinar el número al que hay que llegar.
    Ahora que tienes la solución, borra los números del cuadrado latino original, y deja los resultados y la operación de cada región. Si tu puzzle tiene una solución única, ¡tienes un juego de KenKen!
    Problema D: Envíanos tu mejor puzzle de KenKen original
    Si te gustó jugar al Kenken, envíanos tus soluciones a alguno de los problemas, A, B, C o D. Agradeceremos tu participación en futuros Blogs.

  • My puzzle column in Spanish!

    I’ve started a monthly column for La Voz, a wonderful Spanish language magazine focused on news and culture. La Voz intends to be a meeting place between the Spanish and English speaking communities in the mid-Hudson Valley, and I hope to throw mathematics into the mix!

    My columns include a discussion of interesting math topics, and original KenKen puzzles. I welcome feedback, including solutions and ideas for future topics.
    Here’s a link to my latest article: September 2009
    You can find my older articles in the La Voz Archives.