Category: Class Openers

As students entered today, each group found on their tables the mystery envelope…{cue Law and Order music}.

Inside each evenlope were 36 cut out pieces of paper.  Yesterday we had discussed sample spaces, theoretical and experimental probability involving 2 dice, using applets and online resources to think about similarities / differences, so these pieces were not unfamiliar.  Instructions for the pieces was provided on the board:

Arrange your pieces into a 6×6 grid, so that no duplicate red die appears in any row or column, nor any duplicate white die.

Groups immediately gravitatied towards this task, and after some initial misunderstandings over the directions, got down to business looking for patterns. One student who was a Sudoku fan became quite obsessed. I let this problem hang through the period, just having them stop when we completed notes or had class problems to do.  Many students felt they were close, but couldn’t quite solve the puzzle. Half-way through the period I reminded students that it is often helpful to think about a smaller problem before tackling a larger one. What if you just looked at the dice numbered 1 through 4 – can you complete that problem?  Learn from smaller steps – then tackle the big ones.

If you want to try this task on your own, cut out the pieces from the picture above. Then come back here and visit for the solution.

OR if you just want to know how it end….you unadventurous soul…then keep reading…

We’ll wait for you to come back.

THE BIG REVEAL

The problem I gave today is an interpretation of the 36 Officers Problem, a problem often credited to Euler.  Instead of dice, the Officers challenge is to arrange 6 ranks and 6 regiments into rows and columns.  My alma mater, Muhlenberg College, hosts a high school math competition every February – with a free t-shirt to all who attend. For the 36th year of the contest, the shirt features the 36 Officers Problem, and gave me inspiration for today’s activity.

As for the solution….here’s the thing….the 36 Officers Problem is impossible.  It’s do-able for all other numbers of rows and columns (except 2), but is impossible for a 6×6.  Some examples of solvable squares appear on this untamed blog, and I’d love to have others contribute their thoughts and or/resources.

Classes today were presented with a multiple-choice question as they found their seats –

This summer, Mr. L met a math teacher who has won competitions involving:

A) Reciting digits of Pi

B) Multiplying 5-digit numbers quickly

C) Drawing perfect free-hand circles

D) Finding really large prime numbers

Only one of the above choices is correct, and the class voted on which one they believed to be true. The “pi” option received the plurality of votes.  The video below then reveals the correct answer –

And this is how my classes “met” Alex Overwijk, a World Freehand Circle Drawing Champion, and a fine math teacher I had the opportunity to meet at this summer’s Twitter Math Camp.

After some lively discussion over how one apires to be a proficient circle-maker, and why a World Championship even exists for it, some deeper math ideas emerged, in particular…

How would we judge the roundest circle contest?

One student suggested using a compass to make a similar, perfect circle. Piggy-backing, another student thought we could then somehow measure the “white space” between a drawn circle and a perfect circle?

But what radius should we use? And how do we determine the center of an imperfect circle? So many great questions. To close the opener, I shared how calculus – a few years  down the road – will provide some structure for finding unusual areas.  Always fun to provide some math seeds for experiences down the road.

Many days, I just like to have something interesting on the projector as kids walk in, hoping to see what conversation is generated. This one comes courtesy of Derek Orr.

As students walked in, many immediately became interested in the equations, and found them “neat”.  For others, it took a few minutes until the “ohhhh” moment hit them.  Some highlights from the discussion?

ME: Who do you think figured these out?

Somebody lonely with a lot of time on their hands

STUDENT: Does it matter if we switch the numbers around?

This led to a review of the commutative property, and we quickly realized that switching the numbers was futile here.

My last period of the day became focused on whether there may be other trios with this unique property. I don’t see any real rhyme or reason to the numbers here, and while the kids thought I might be playing “dumb” I really have no clue if there may be many more examples, or if this is all there is.  But then came the magic words kids of this generation hug like blue blankies:

Let’s Google It!

I was pretty skeptical that anything would come from a Google search.  What would you google anyway? How do you phrase it? But sure enough, after a few minutes one of my students found references and some more interesting examples:

How cool is that!  More on these intersting number facts can be found in a paper called “A Curious Cubic Identity and Self-Similar Sums of Squares” .  Check it out and the associated research!