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A11. Splitting a triangle and a tetrahedron

János Kurdics published this interesting problem in The Math Connection Linkedin group:
Given a triangle, find the shortest line segment that divides the triangle into two regions of equal area. A solution that does not involve calculus is preferred.
The solution that I know is very nice, and gives quite a workout in elementary trigonometry.

János says that he is working on three-dimensional analog (plane that divides a tetrahedron into two regions of equal volume and gives smallest possible cross-sectional area with the tetrahedron) but has only solved it for a regular tetrahedron.

A10. Triangle with sides in arithmetic sequence.

Find a number n such that there is a triangle with sides n, n + 1, and n + 2 in which the largest angle is twice the smallest angle. How many such numbers n are there? Note that n does not have to be an integer.

This problem came up in a high school textbook during a tutoring session, except there were several hints given that made the problem much easier. See if you can do it without hints.

E 30. A Cryptarithm

Introduction: Here is a problem that I think would be good for any child who knows how to use the standard algorithm to do multi-digit multiplication. It is a cryptarithm from The Moscow Book of Puzzles, an amusing book by Boris Kordemsky, available in a very inexpensive Dover edition. For those of you who don't know (as I didn't) a cryptarithm is like a cryptogram, except the unknowns are digits, not letters. One you may have seen is (S  E  N  D) + (M  O  R  E) = (M  O  N  E  Y), where each letter stands for a digit, with different letters standing for different digits, and the expressions in parentheses standing for multi-digit numbers.

Problem: In the following multiplication, each * stands for a prime number digit (2, 3, 5, 7). Replace each * by a prime digit so that the multiplication is correct.

      * * *
     x  * *
   * * * *
 * * * *
* * * * *
Strangely enough, you have all the information you need to find the unique solution.

Multiplication table problem

Sometimes mathematics problems that are interesting to the professional mathematician can arise out of the simplest questions. We all learned a multiplication table in school, which lists all 100 products of integers from 1 * 1 to 10 * 10. I've been told that in England, students are required to memorize up to 12 * 12. In the multiplication table for 1 to 10, there are 100 products, but it is only necessary to learn 55, because of the commutative property. In other words, we only need to learn the products a * b, where a ≤ b, and the number of such products is 1 + 2 + ... + 10 = T(10) = (11)(10)/2, where T(n) is the n-th triangular number, (n + 1) * n / 2. Therefore, the number of different products is at most 55. Actually it is less. For example, 6 appears twice in the triangular table, as 1 * 6 and as 2 * 3. The number of different products in a 10 by 10 multiplication table is 42. (Shades of Douglas Adams!)

A mathematician would naturally be interested in knowing how many different products there are in an n by n multiplication table, in other words, the number of different products of the form ab, where a and b are positive integers less than or equal to n. Call this sequence a(n). Then a(n) ≤ T(n), so the lim sup of a(n)/n^2 as n goes to infinity is less than or equal to 1/2. In fact, Erdös gave a very nice proof that the limit is 0, and he and others obtained more accurate asymptotic formulas for a(n).

See the Online Encyclopedia of Integer Sequences, where a(n) is given as sequence A027424. Links provide much information about the sequence. Also, the question of how many different numbers appear in a multiplication table could be given to students at almost any level.

E29. Divisible by Sixteen

Here's an easy one:

You have sixteen balls, numbered consecutively 1 through 16, and 4 boxes. You are asked to put 4 of the balls into each box, so that the product of the numbers in each box is divisible by 16. Give a solution, or prove that this cannot be done.

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.