E28. A balancing cube
A sculpture is in the form of a cube, one meter on a side, balancing unstably on one vertex. Let A be the bottom vertex (on the ground), B one of the three adjacent vertices, and C the vertex at the other end of the space diagonal of the cube from A. Assume the ground is a horizontal plane, and AC is perpendicular to to the ground. Find the distance from B to the ground, with answer in exact form.
Alexander Grothendieck
Alexander Grothendieck passed away last month, one of the giants of twentieth century mathematics. Victor Gutenmacher sent me a link to an article by Pierre Cartier celebrating Grothendieck's life. I am somewhat chagrined that I knew so little about Grothendieck prior to reading the article, which I highly recommend.
Someone once said that there are two types of first-rate mathematicians: problem solvers and theory builders. Grothendieck was one of the latter. He left tens of thousand of pages of work, and directed a brilliant group of researchers at the Institut des Hautes Etudes Scientific who carried forward the program that he developed.
A grand synthesist, Grothendieck worked in fields as seemingly diverse as functional analysis, algebraic geometry, group theory, homological algebra, and Galois theory.
The remembrance of Grothendieck is "A Country Known Only by Name", a title that makes sense after reading the article. The author, Pierre Cartier, was a friend and colleague of Grothendieck and writes about both the man and his work. Grothendieck was principled to the point of eccentricity, and seems to have been very difficult to get along with. After a dispute over military funding in 1970, which offended his pacifism, he retired from involvement in the mathematical community and lived from 1988 in isolation.
This article is at http://inference-review.com/article/a-country-known-only-by-name. It is lengthy, but in my opinion well worth the time.
Someone once said that there are two types of first-rate mathematicians: problem solvers and theory builders. Grothendieck was one of the latter. He left tens of thousand of pages of work, and directed a brilliant group of researchers at the Institut des Hautes Etudes Scientific who carried forward the program that he developed.
A grand synthesist, Grothendieck worked in fields as seemingly diverse as functional analysis, algebraic geometry, group theory, homological algebra, and Galois theory.
The remembrance of Grothendieck is "A Country Known Only by Name", a title that makes sense after reading the article. The author, Pierre Cartier, was a friend and colleague of Grothendieck and writes about both the man and his work. Grothendieck was principled to the point of eccentricity, and seems to have been very difficult to get along with. After a dispute over military funding in 1970, which offended his pacifism, he retired from involvement in the mathematical community and lived from 1988 in isolation.
This article is at http://inference-review.com/article/a-country-known-only-by-name. It is lengthy, but in my opinion well worth the time.
A9. Squares erected on a triangle.
Here is a nice problem from Coxeter and Greitzer’s classic book,
Geometry Revisited, where an elegant
solution is given.
Let ABC be a triangle, and construct squares externally on
the sides. Let O1 be the center of the square on AB, O2
the center of the square on BC, and O3 the center of the square on AC,
as in the following diagram:
Prove that the segments O1O3 and O2A have the same length and are perpendicular to each other.
Multiplication & Division of Polynomials
When I was in high school I learned to multiply polynomials by using a technique that was reminiscent of the multiplication of multi-digit numbers that I had learned in earlier grades. That is, after filling in any "missing terms" with zeros the two polynomials were written one over the other, with the terms arranged by degree, with the constant term at the right. If the polynomials were of degrees m and n, with m <= n, the polynomial of degree m was written on the bottom. We would multiply the top polynomial by each of the m terms of the bottom polynomial, to produce m rows, each of which was indented one position to the right. Then all terms of the same degree were located above one another, and we would add each column to get the answer. Similarly, the algorithm for polynomial long division was based on the algorithm for multi-digit integer division. Polynomial division never really clicked for me. I learned how to do it, but somehow it felt like magic.
These days, most elementary and middle school students - even many students from better schools - are not being taught multiplication and division of multi-digit numbers, instead being told they can rely on the calculator app on their smartphone. This has caused much hand-wringing, particularly among traditionalists. How can we teach students to multiply and divide polynomials when they haven't learned to multiply and divide integers? Non-traditionalists might say that students can use CAS to multiply and divide polynomials and don't need to be able to do it by hand, but I am not willing to go that far.
In the course of tutoring a high school student, I recently was exposed to a method for multiplying polynomials that is being taught currently in some schools, called the galley method . I really like it, for several reasons:
(1) The method does not not depend on knowing the analogous arithmetical algorithm.
(2) The method is directly related to the definition of the multiplication of m and n as the area of an m x n rectangle (or as the cardinality of the Cartesian product of sets of cardinalities m and n, if you prefer). These definitions are, in the opinion of many educators, superior to regarding multiplication as repeated addition.
(3) The first time I used the galley method, I found it slightly easier to use than the traditional method. I would use it myself in cases where a CAS is not necessary or available.
(4) The galley method immediately leads to the reverse galley method for the division of polynomials.
Here's how it works: To multiply polynomials of degree m and n, create an (m+1) by (n+1) table, the galley. (The term galley apparently comes a French word that indicates an oblong tray for holding setup type.) Write the terms of the degree n polynomial above the n+1 columns, in order of decreasing degree as you move from left to right. Write the terms of the other polynomial along the right side of the galley, in order of decreasing degree. Then, fill in each cell with the product of the monomials above its column and to the right of its row. Finally, observe that the terms on any diagonal (of slope 1) have the same degree, and the degrees corresponding to different diagonals are different. Add along these diagonals to combine like terms, and write the result in the first empty space outside the galley. The result is now in front of you, in an L-shaped pattern along the left and bottom sides of the galley. If you find this difficult to visualize, you can find an excellent description with diagrams here.
The galley method can be used in reverse for polynomial division. Suppose you want to divide a polynomial of degree m by a polynomial of degree n, where n < m. Create a galley of n+1 rows and m-n+1 columns. Write the n+1 terms of the divisor along the right side of the galley and the m+1 terms of the dividend in an L-shaped band beginning in the area one unit to the left and one unit below the upper left-hand corner of the galley. Then fill in the terms in the remainder of the galley as well as the terms above the galley one by one, beginning with row 1 column 1 of the galley, then row 0 column 1 (just above row 1 column 1), then the rest of column 1. Next move to row 1 column 2, and so forth. The solution will appear above the galley. If there is a remainder, some of the last n-1 terms in the divisor (the L-shaped band) will not in fact be the sum of the terms on their diagonals, and the remainder is the difference between the sum of the terms on the diagonals and the corresponding terms in the divisor. Again, if you need further description, go here.
These days, most elementary and middle school students - even many students from better schools - are not being taught multiplication and division of multi-digit numbers, instead being told they can rely on the calculator app on their smartphone. This has caused much hand-wringing, particularly among traditionalists. How can we teach students to multiply and divide polynomials when they haven't learned to multiply and divide integers? Non-traditionalists might say that students can use CAS to multiply and divide polynomials and don't need to be able to do it by hand, but I am not willing to go that far.
In the course of tutoring a high school student, I recently was exposed to a method for multiplying polynomials that is being taught currently in some schools, called the galley method . I really like it, for several reasons:
(1) The method does not not depend on knowing the analogous arithmetical algorithm.
(2) The method is directly related to the definition of the multiplication of m and n as the area of an m x n rectangle (or as the cardinality of the Cartesian product of sets of cardinalities m and n, if you prefer). These definitions are, in the opinion of many educators, superior to regarding multiplication as repeated addition.
(3) The first time I used the galley method, I found it slightly easier to use than the traditional method. I would use it myself in cases where a CAS is not necessary or available.
(4) The galley method immediately leads to the reverse galley method for the division of polynomials.
Here's how it works: To multiply polynomials of degree m and n, create an (m+1) by (n+1) table, the galley. (The term galley apparently comes a French word that indicates an oblong tray for holding setup type.) Write the terms of the degree n polynomial above the n+1 columns, in order of decreasing degree as you move from left to right. Write the terms of the other polynomial along the right side of the galley, in order of decreasing degree. Then, fill in each cell with the product of the monomials above its column and to the right of its row. Finally, observe that the terms on any diagonal (of slope 1) have the same degree, and the degrees corresponding to different diagonals are different. Add along these diagonals to combine like terms, and write the result in the first empty space outside the galley. The result is now in front of you, in an L-shaped pattern along the left and bottom sides of the galley. If you find this difficult to visualize, you can find an excellent description with diagrams here.
The galley method can be used in reverse for polynomial division. Suppose you want to divide a polynomial of degree m by a polynomial of degree n, where n < m. Create a galley of n+1 rows and m-n+1 columns. Write the n+1 terms of the divisor along the right side of the galley and the m+1 terms of the dividend in an L-shaped band beginning in the area one unit to the left and one unit below the upper left-hand corner of the galley. Then fill in the terms in the remainder of the galley as well as the terms above the galley one by one, beginning with row 1 column 1 of the galley, then row 0 column 1 (just above row 1 column 1), then the rest of column 1. Next move to row 1 column 2, and so forth. The solution will appear above the galley. If there is a remainder, some of the last n-1 terms in the divisor (the L-shaped band) will not in fact be the sum of the terms on their diagonals, and the remainder is the difference between the sum of the terms on the diagonals and the corresponding terms in the divisor. Again, if you need further description, go here.
E27. A puzzling sequence
Someone gave me the following puzzle at a party. I don't know the source. It involves nothing beyond elementary school mathematics.
Find a simple pattern in the following sequence, and give the next row:
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1
Find a simple pattern in the following sequence, and give the next row:
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1
E26. Final digit of a power
The following is adapted from Posamentier and Salkind's Challenging Problems in Algebra, published by Dover Books.
(a) Prove that for any positive integer n, the final digit of n5 is the same as the final digit of n.
(b) Without using a computing device, find all positive integers whose 5-th power is a 7-digit number that ends in the digit 7.
(a) Prove that for any positive integer n, the final digit of n5 is the same as the final digit of n.
(b) Without using a computing device, find all positive integers whose 5-th power is a 7-digit number that ends in the digit 7.
Don't Be a Math Teacher
I have enjoyed teaching math for several decades and
currently teach K–12
math teachers about education and mathematics. So you'd think that I would
recommend that a mathematically-inclined person should give strong
consideration to becoming a math teacher. You'd be wrong.
If when I began my career the state of the mathematics
teaching profession, both at the school and university levels, was what it is
today, I don't think I would have gone into it. Math teaching now has the
following difficulties.
First, digital technologies have resulted in students with
shorter attention spans and more dependence on instant gratification. You would
be surprised how many high school students are unable to add or multiply two
two-digit numbers without a calculator, and in many cases unable to add or
multiply even one-digit numbers. This matters, in much the same way that it
would matter if students couldn't read words of more than one or two syllables,
and depended on print-to-voice readers to “read” literature. Also, most students faced with a problem
that requires more than five minutes of work, most will say that they can't do
it. A Korean student that I had in a Calculus course in an elite U. S. woman's
college earned an A+ told me privately that she felt she was not good at
mathematics but was best in the class because the rest of the girls were lazy.
I think we will only see more and more students who are unwilling to do the
hard work required to become good at math.
Second,
in the United States, virtually all K–12 students must pass tests in order to
advance in grade or to graduate. Teachers must teach to the test. While
teaching to the test is, in my opinion, a good thing for formative assessments,
teaching to the test for these high-stakes one-size-fits-all tests at best
limits the ability of the teacher to approach the curriculum in innovative ways
and at worst results in much time wasted with students taking practice tests
and learning test-taking tricks that have nothing to do with mathematics. And
of course, if the students do not do well on these tests, the teacher's job is
on the line.
Third,
computers are in the process of making math teaching obsolete. If a student can
learn a subject online, where they can have their instruction completely
personalized and can have their homework graded instantly, why should a school
system invest in expensive and inefficient human beings to do the teaching?
Most teachers will be replaced by low-paid “facilitators” of online courses who
do not need to even be trained in mathematics. Instead of there being thousands
of college instructors nationwide teaching Calculus I, a dozen or so master
teachers will make videos and consult with courseware designers to put together
online courses. Math teaching, as we know it, will exist only at the graduate
school level where economies of scale do not apply, or at pricey private
schools.
So my career
advice is only to become a math teacher if it is really a calling for you. Good
teaching will be harder than it has ever been, with fewer rewards and little in
the way of job security.
Fun with divergent series
A friend sent me a link to a Numberphile presentation in which Tony Padilla and Ed Copland show that the infinite series S = 1 + 2 + 3 + 4 + ... has (in some sense) the value - 1/12. It seems that this result is used in superstring theory in physics and goes all the way back to Euler in 1740. It has generated a lot of heat (and some light) on one of the math forums I follow, so I looked into it and have written it up a bit.
E 25. Dividing cards
Here is a problem from Math Charmers, a delightful book by Alfred S. Posamentier.
You are in a completely dark room, seated at a table. There are twelve playing cards laid out in front of you. You know that five of them are face up and the remaining seven are face down, but you have no way of determining whether a card is face up or not. Your task, should you decide to accept it, is to divide the cards into two piles so that each pile has the same number of cards face up as the other. You may, of course, turn cards over.
While at first this sounds impossible, there is a simple solution.
You are in a completely dark room, seated at a table. There are twelve playing cards laid out in front of you. You know that five of them are face up and the remaining seven are face down, but you have no way of determining whether a card is face up or not. Your task, should you decide to accept it, is to divide the cards into two piles so that each pile has the same number of cards face up as the other. You may, of course, turn cards over.
While at first this sounds impossible, there is a simple solution.
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