(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. (S*ee Newton and Leibniz)*