## Saturday, October 5, 2013

### Playing With Math - The Book

I've been working on Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers for almost five years now. For years I've been saying "It's almost done." I have sometimes felt bad about how long it has dragged out. I'm not sure if the authors (over thirty of them!) trust that the book will really get into print. It will.

Last week I was perusing The World of Mathematics, by James Newman (a lovely four-volume mathematical encyclopedia), so I could write a description of it for my Book Picks section in Playing With Math. He mentioned in the preface that it took him fifteen years to put it together! Ok, now I don't feel so bad.

I've been sending out little updates about my progress on my facebook page for Playing With Math, and mentioned that there today. If you'd like to follow my progress, please 'like' that page.

Now I have something slightly bigger to say, which seems to fit better here on my blog. I'm working on answers to the puzzles Paul Salomon created - his Imbalance Problems. I am stuck on the last one we included. It may be that my brain is fried, and you'll all see an answer easily. But I just don't.

This puzzle was created by one of Paul's fifth grade students, Felix. Can you solve it?

Now I wish I had used these at the beginning of our inequalities unit in pre-calc. They really make you think about what > and < mean! I think I'll hand them out anyway - as a challenge.

1. This reminds me of some of the puzzles from the US Puzzle Championship. (e.g. puzzle 10 on the 2002 test: http://wpc.puzzles.com/history/tests/qtest2k2/index.htm)

One useful strategy is to use actual numbers to satisfy the constraints. For instance, one can start on the right side where 1 circle is heavier than 2 circles + 1 square. Choose values for the circle and the square that satisfy that constraint, and then come up with a value for the triangle that makes the global constraint true, and then the relationships between the specific values should be a valid set of relationships. (And if the answer is known to be unique, then you have found *the* unique answer rather than one possible answer.) It's more difficult to apply this method if the you can't approach the problem just by looking at local constraints. (For example, if there were some other constraint in the puzzle above that imposed more restrictions on the values of circles and squares, and therefore both constraints need to be used together to come up with possible weights.)

2. I think this one is hard because it breaks the physical intuition of mobiles, where the vertical connections are typically some kind of string. In that setting, negative weights (helium balloons perhaps) don't make much sense, because the corresponding objects might float above the rigid cross bars. But the right half of the diagram demands that either circles or squares or both be negative. For exampe, you can make the whole thing work with circles as -3, but then the rightmost circle would be drawn wrong. Alternatively, you can make the whole thing work--including all the drawings--with squares as -3 (assuming you allow fractions). The price, however, is significantly more constraints on the problem, because you must make sure that every "suffix" of every dangling chain is positive.

1. (Chris, thanks for the comment. This got stuck somehow on blogger.)

3. @Hao-
There's a whole list of those torque-based puzzles here:
http://www2.stetson.edu/~efriedma/weight/

As for your method, I think it certainly *can* get you a solution, and for most, somewhat simple puzzles, it is likely to work, but it depends on your initial selection of values. Depending on your choice, you may or may not be able to find a third weight to solve the puzzle at large. Does that make sense?

4. Sue! I've never seen these puzzles - they are so awesome. I just pinned it to pinterest and am fixing to print them to work.

Thanks for sharing them!

5. And my thanks to Paul for sharing his creation!

6. Psst: check out http://solveme.edc.org

7. Bowen's link takes you to a page with puzzles that are somewhat similar to Paul's. Bowen, do you work with the folks who made that page? Can you tell us more about why you recommend it?