There have been attempts to add some modern work to the curriculum – such as graph theory – but even in those topics the knowledge of the basics goes back many years. Thus mathematics can be perceived as ossified; this can even affect teachers who see no need to update material. Applied work can give some semblance of modernity, but still in the end pure mathematics seems unchanging.

In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1.

-- From the abstract for Kuelshammer ideals and the scalar problem for blocks with dihedral defect groups by Thorsten Holm and Guodong Zhou

Wouldn’t it be nice for students to experience at least a little of the recent mathematical world? Perhaps just maybe, in the process, solving a problem that has never been solved before?

Case study: Chess problems

(Note this section assumes a familiarity with the movement of chess pieces.)

Recreational mathematics (defined as the study of mathematical puzzles and games) is one area where the fringes are accessible even at the elementary level. This is partly due to the relative youth of the discipline, partly due to it being easy to find an unexplored area (just pick a new game or puzzle), and partly due to the students being able to do research by quite literally playing a game.

Erich Friedman’s Math Magic column had an article in February of 2007 about a particular kind of chess puzzle:

This month we investigate chess positions containing two types of pieces A and B subject to the condition: each A attacks exactly n B's (and no A's), and each B attacks exactly m A's (and no B's). Can you find smaller solutions? Can you solve one of the unsolved cases? What are the smallest solutions for other pairs of pieces?

-- Erich Friedman

What intrigued me about this column was the preponderance of question marks. There’s a table which gives solutions for every combination of chess pieces, and some answers are unknown.

To simplify things for students, I focused on the problems with a “collections of pieces (possibly equal) where each piece attacks exactly 2 of each type of piece, including its own.”

For example, in the grid below, each king attacks two other kings and two rooks, and each rook attacks two other rooks and two kings.

Note how these cases are (as of this writing) completely unknown:

• Bishops and Knights
• Kings and Knights
• Queens and Knights
• Rooks and Knights

Could they be solved by students? Could they prove any are impossible?

In Practice

I have attempted this in an actual classroom. I started with the 8 queens puzzle: Can you fit 8 queens on a standard 8x8 chessboard so no queen attacks each other? I then moved on to demonstrate some of the problems from Erich Friedman’s table, and picked a few more for the class to solve. Finally, the students attempted to fill in the unknowns.

Unfortunately the experiment was a failure, and I have a few notions why.

1. I didn’t have enough scaffolding leading up to the unknown questions. The 8 queens puzzle and a handful of examples weren’t enough to give the students a real feel for how the chess puzzles worked. They also needed more successes to build confidence before they got on to the real puzzlers.

2. There is the possibility the empty spaces on the chart have no solution. (Bishop-King and King-King have now been proved impossible, but were still open problems when I did this with my class.) If there is an answer the students simply have to demonstrate it, but if there is no answer the students have to demonstrate a proof of why not. The students never had any practice with this (at least in the context of chess puzzles).
   

3. Some students seemed to cue on to this being an “enrichment” activity and decided to zone out until we returned to the “real” curriculum.

4. There just wasn’t enough time in class. Students needed their own pieces at home to really be able to work.

Projected Solutions

I may attempt this experiment again. Here are my proposed solutions to the issues:

1. Add more examples for the students to try on their own and in groups to build confidence.
   

2. Go through at least one proof of an impossible position. Proving that it is impossible to lay out kings such that each king attacks exactly two other kings is within the range of students, and I plan to dedicate enough time so the students get a better sense of the proof method involved.

3. Put something from the activity on the next test, forcing the students to realize this isn’t just a fun side digression.
   

4. Make paper sets that the students can use at home.

Further Explorations

While I presented the chess problems with a hope students might solve them, it’s possible to use problems the students have no chance of solving as a curriculum anchor. I had good success with a lesson around one of the Millennium Problems, which I’ll get into next week.

Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.

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