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.”

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)


19 responses to “Imbalance Problems

  1. Did you draw those by hand?

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  3. Doesn’t problem 2 have three possibilities? My daughter and I thought the shapes in question could be equal, or either of them could be the heaviest.

    • Hi Denise- You’re right. Problem 2 is not well-written. It has multiple solutions, as you described. Triangle and Square are both heavier than Circle, so Circle is the lightest, but it’s unclear who’s heavier between the triangle and square. I fixed the problem in my new post, and I added many more. I hope you get a chance to try those as well. #6 is especially tricky. Thanks for commenting.

  4. We did these for school today. Lots of fun!

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  10. William Moates

    So, each shape has a uniform weight? (All triangles are the same weight, all squares are the same weight, etc.) In the sketch above, I assume those are only squares, and not “squares and rectangles”.

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  14. Hey Paul — Do you have solutions to these that I can share with my students? Thanks!

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