# Solving Math Problems

## Starting

Starting in december 2001 with Problem 26 from the NAW I became interested in problemsolving. Such a simple formulated problem:

'Does there exist a triangle with sides of integral length such that its area is equal to the square of the length of one of its sides'.

Learning about elliptical curves and other fine topics of Number Theory I found a solution, and an other one, one more, etcetera. You can find the published solution in the NAW 5/3 nr. 3 and my collection of solutions can be found here ( .ps, .pdf). Recently (july 2005, ok not so recently) I found solution number eight, based on a representation of Heron triangles found in my solution of UWC Problem2005-1C.

## Next Problem

The next problem that took me by surprise was Problem 29 of the NAW. There was an erroneous solution in NAW 5/3 nr. 3 and the problem was declared open again in NAW 5/3 nr. 4. The editor of the problem section challenged me to attack this unsolved problem. I found a solution in the beginning of Januari 2003. Interesting is to know that Problem 29 originated from work related to a paper of Lute Kamstra: Juggling polynomials, CWI Report PNA-R0113, July, 2001. Solutions of Problem 29 can be generated with my C-program problem29.c.

For myself I translated and extended the problem to a Dancing School Problem: How to match boys and girls in a dancing class under certain length restrictions. My story of the Dancing Schools includes the solution of Problem 29, but also links with certain kinds of Rook Placing Problems. There is a SAGE-program to generate polynomial solution to a certain class of problems. From the Dancing School Problem originated the sequences A079908-A079928 from the OEIS (see below).

My solution of problem 29 is in terms of the permanent of (0,1)-matrices. So I became interested in Permanents. Playing with Maple and counting I found an alternative for the famous Ryser's algorithm. I implemented my algorithm in a C/C++program, which was used to contribute to Neil Sloane's On-line Encyclopedia of Integer Sequences (OEIS). See for instance A087982, A088672 and A089476.

The dancing school returned in a problem published at Elsevier's www.mathematicsweb.org (this link was in abuse) as problem 3 (yet unsolved!?). The link to Elsevier's Problem Section seems to be dead.

Problem 29 once again showed up in disguise as part 2 of Problem2006-2B (see below) in the NAW 5/7 nr. 2. There were no solutions sent in, so this is a waste of a nice problem!

There is an article in the NAW 5/7 nr. 4 December, 2006: Dancing School problems, Permanent solutions of Problem 29. A preprint can be found here.

## UWC/Problems

The problem section of the NAW was discontinued and merged with the UWC, the University Math Competition, open for Belgian and Dutch math students. Starting with NAW 5/4 nr.1 the Problem Section and the UWC became the section Problemen/UWC. First in Dutch, but later on my suggestion the problems are formulated in the English language, as are the solutions. Students can gather points with their solutions. Others can send their solutions 'hors concours'.

 NAW5/4 nr. 1 opgave2003-1A NAW5/4 nr. 3 opgave2003-3A, opgave2003-3B, opgave2003-3C NAW5/4 nr. 4 opgave2003-4A, opgave2003-4C NAW5/5 nr. 1 opgave2004-1A, opgave2004-1B NAW5/5 nr. 2 opgave2004-2A, opgave2004-2C

In the NAW 5/5 nr. 3 there was no UWC/problems section, due to a misunderstanding between the editorial board and the editors of the section. There was a change of editors starting with the NAW 5/5 nr. 4. Note the difference: Opgave is replaced by Problem

 NAW5/5 nr. 4 Problem2004-4A, Problem2004-4B NAW5/6 nr. 1 Problem2005-1A, Problem2005-1C NAW5/6 nr. 2 Problem2005-2A, Problem2005-2C NAW5/6 nr. 3 Problem2005-3B NAW5/6 nr. 4 Problem2005-4C

A new serie of problems:

 NAW5/7 nr. 1 Problem2006-1B NAW5/7 nr. 2 Problem2006-2B NAW5/7 nr. 3 Problem2006-3C NAW5/7 nr. 4 Problem2006-4C

The UWC has now changed back into a general Problem Section, open to everyone.

 NAW 5/8 nr. 1 Problem2007-1A, Problem2007-1B NAW 5/8 nr. 2 Problem2007-2A NAW 5/8 nr. 4 Problem2007-4C

The problems of the year 2008.

 NAW 5/9 nr. 1 Problem2008-1A NAW 5/9 nr. 2 NAW 5/9 nr. 3 Problem2008-3A, Problem2008-3B NAW 5/9 nr. 4

The problems of the year 2020.