Imbalance Problems

(Click here for full page of imbalance problems.)

My fifth graders have been writing problems this week.  Mostly things liks “5 branches have 40 leaves, and 10 trees have 200 branches.  How many leaves will 320 trees have?”  Some of them have been writing problems with symbols that amount to systems of linear equations, and earlier this year we worked on balance problems to get at that sort of thing.

While all this was going on, inspiration struck.*  What if the scales don’t balance?  What if one side weighs more?  Behold, my “imbalance problems.”

Unfortunately, problem 2 cannot be uniquely deduced.  It has two possible solutions.

These are just three I came up with for example, but I can see a whole world of possibilities.  I love how designing them is maybe a better problem than solving them.  How many pieces of information do I need to give (albeit, implicitly)?  [turns out it depends on what the information is! (maybe that’s not surprising.)]

Imbalance Problem Contest

Inspired by this post by the incomparable Shawn Cornally, I’ve decided to offer a little puzzle-writing competition.  Spend some time writing great imbalance problems (the kind that push the state of the art), then share a link to them in the comments.  My two favorite puzzle-writers (based on indescribably subjective criteria) will win a print of their choosing from my Stars of the Mind’s Sky series, up to 12″x12″.  All of the rest of us will get a collection of excellent imbalance problems to solve and share with others!  Exciting right?  Who’s in?

(Click here for contest results)

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*not a ground-breaking, revolutionary, or truly novel inspiration, I admit.  For me, this is a clear cut Cohen-Ventorism.**  It’s such a slight tweak on an extremely common problem type.  Someone else (countless others) must surely have also come up with this.  And yet, a google search for “imbalance problems” yielded surprisingly little.

** Cohen and Ventor are two imaginary mathematicians who completely independently, perhaps simultaneously, discover (or coinvent) all sorts of mathematics.  (See Newton and Leibniz)

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My New Art Gallery (and a bonus)

I’ve been very inactive on this blog for a while now (though a lot of great things are happening over at Math Munch (big news to come…)), but I have actually been very active.  This blog has been mostly for math/school pontification, but I’ve decided to start posting more broadly, beginning with more mathematical art stuff.  To that end, I’m creating an art gallery of my works here on Lost in Recursion.

You can click on the Art tab at the top of the page, or you can hover for more specific pages.  I hope you’ll click around, because I’ve included lots of images and little descriptive blurbs.  I’ll post a few pics below to whet your appetite.  Click for more of the same.

I’ll continue to add to that section and use it to document and publish my work as an artist.  Speaking of which, I’m very excited to be included in the Saint Ann’s Faculty and Staff Biennial Art Show.  More on this another day.  I’ve also gotten access to a couple of very nice printers which can go as wide as 13″, so I’m excited to start making some beautiful prints of these images I’ve had lying around a while.

Here’s your bonus: a fantastic video that Justin put together showing off the three sizes of zome nodes used in our meta-meta-node.  Enjoy it all and leave a comment, why don’t ya!

Stars of the Mind’s Sky

I just submitted some art for exhibition at Bridges 2013, the world’s largest mathematical art conference.  I’m extremely hopeful that it will be accepted.  I actually submitted art for exhibition at the Joint Mathematics Meetings, but it was denied, I think, because I hadn’t figured out how to get the right image size and resolution.  Whoops.  Anyhow, I’m proud of what I put together, and I wanted to share it with you here, so I’ve pasted the art, description, and my artist’s statement below.  (And yes.  I will sell you a print if you’d like.)

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My earliest love of mathematics came when I realized its utility in creating beautiful things. As a third grader (and ever since), I spent countless afternoons drawing stars and patterned shapes with a protractor or compass. In my work as a teacher I get to share that beauty in creation with my students. In my art I sometimes try to illuminate the complex structure and interconnectedness of simple, patterned objects. I’m compelled to understand complete spaces of related works and how my choices as an artist locate me within that space.

Stars of the Mind’s Sky
22.75 in x 22.75 in
digital print
2012

Stars of the Mind’s Sky is the title of a series of works exploring the space of regular star polygons. Here we see 300 stars “in orbit” along concentric circles. The number of points on a star increases with the radius, and stars of a given number of points are spaced evenly along their circle according to “density,” or the “jump number” used in generating them. Algebraically, these represent the subgroups and cosets generated by elements of a cyclic group. They have been colored on a gradient to indicate the number of cosets; a red star signifies a generating element. As a consequence of these structural choices, we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.