### Polyhedral Crafting

I started the year once again feeling unsatisfied with the spare, utilitarian look of my classroom.  So, having given up on finding math-related posters I liked, I decided to head over to Math Monday to look for some cool looking thing I could make this weekend.  The result:

Tensegrity polyhedra, made from 3/16″ dowels and standard rubber bands, based on this little article by George Hart.  The coolest thing about them is that no two sticks are actually touching each other (which makes me wish I’d used different color rubber bands).  Or maybe it’s that they can collapse like this…

…and then snap back into shape.  Or maybe that they bounce. (Yep.)

In any case, they were really fun to make–the dodecahedron turns out to be a great spatial reasoning puzzle as you get close to the end–and I think they’ll make good toys or decorations.   And there’s tons more inspiration for mathy crafts at Math Monday (as well as at georgehart.com and vihart.com).  Maybe I should crowdsource this by offering it as extra credit–that ought to get the classroom looking good in no time :)

### Postgame Analysis: the Towers of Hanoi

I recently gave my juniors the classic Towers of Hanoi puzzle to play with in small groups.  It went something like this:

You have three plates, and plate #1 has a stack of 5 pancakes, in order from the largest one on the bottom to the smallest on top.  The puzzle is to get the stack onto plate #2 using as few moves as possible.

Two rules: (i) you can only move the top pancake on a stack, and (ii) at no time can any larger pancake be on top of a smaller pancake.

They spent a couple minutes getting familiar with the mechanics of it, and then settled into working together, shifting pancakes and keeping a count of their moves.

### The Problem That Never Fails

[This post also appears as a guest post over at Sam Shah’s blog.]

This is a post about a problem that never fails.  It’s the problem I used for my sample lesson when interviewing for jobs four years ago.  It’s the one I almost always use on the first day of class, and it’s also what I give to parents on back to school night.  Because… well…it. never. fails.  Seriously.

The perfect, ineffable jewel of a problem to which I refer is the classic Bridges of Konigsberg problem.   Here’s the story, in case you don’t know it:

(image from wikipedia)

As shown in the image above, the town of Konigsberg once had seven bridges.  Back before some of these bridges were bombed during WWII, the residents of the town had a long-standing challenge: to walk through the town in such a way that you crossed each bridge exactly once—i.e., without missing any bridges, and without crossing any of them twice.

So, why is this problem so great for a high school classroom?  Well, first of all, whenever I tell this rather contrived tale to my students (or their parents, for that matter), they are inevitably scribbling on their scrap paper before I can even finish.  It’s a compelling puzzle, simple as that.

Before long, students have redrawn the thing enough times that they’re annoyed with all the extra time it takes to draw all the landmasses and bridges, and so they simplify it:

(image from wikipedia)

Voila: in a completely natural fashion, they have reduced the problem just like Euler did.  At this point, I usually bring them together for a moment to appreciate what’s going on here: reducing a problem to its essential components, finding the simplest way to represent the underlying structure of the situation.  (And I also mention to them that this is precisely the move that Euler made when he invented graph theory based on this initial problem.)  Even if they never went any further, this is already a nice lesson in problem solving.

[SPOILER ALERT: notes on the solution below the fold]

### Mathematical Doodles

Just doodled these “Oroborromean rings”…

… which I learned how to do from the video below. The video is one of the loveliest things I’ve seen in a while… both for the doodles themselves, as well as for mathematically playful, yet also slightly snarky commentary.

### Mice and Wine

This is one of my favorite problems:

You’re planning a huge party for tomorrow, which will include a toast exactly 24 hours from this moment.  You have 1000 bottles of wine, but one of them is contaminated with a slow-acting poison that will kill any living thing within 24 hours of being ingested.  You happen to have 10 altruistic mice on hand who have volunteered to test the poison.  How many bottles of wine can you safely serve at the toast?

I’ve given it to a number of classes, ranging in age and strength, and it’s produced wonderful discussions every time.  Here’s a reconstruction of how many of these have gone.

Right off, several students come up with the idea of splitting up the 1000 bottles evenly among the 10 mice.  When one of the mice dies, they explain, you would know that the poisoned bottle was among the 100 that it drank, and so the remaining 900 would be safe to serve.

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