Too hard

The challenge problem was difficult.  The interesting question is: why?

The teachers in our tutor’s center now and then put up challenge problems on their whiteboards.  These are intended to get the students thinking outside the normal routine and outside the normal lesson plans.  Sometimes they are logic problems, sometimes geometric; mostly they avoid the higher levels of math, since not all students are there yet.  Other teachers generally pick theirs from outside sources, but our tutor recently put up a geometric problem suggested by our astronomer.

No students could solve it (as of two or three weeks later).  Indeed, none of the other teachers has come up with a solution, though one of them (with a degree in mathematics) had an approach using infinite series.  But it hadn’t impressed us as being particularly difficult.  One had to draw some extra lines and run through the algebra, but it was only a couple of steps to set up, and a few more to work through.  No math beyond Algebra 1 and Geometry was needed.  Why did others find it so hard?

[If you really want to know: a circle of radius R is inscribed in a square.  A smaller circle, of radius r, is the largest that can be tucked into the corner between the big circle and the square.  What is the ratio r/R?]

Our tentative conclusion is that our consultants are unusually practiced at analyzing and working with geometric diagrams.  We’ve all spent years drawing things in two and three dimensions, extracting the relevant bits, then doing the resultant math.  Our astronomer is especially comfortable with circles, ellipses and lines tangent to them.  After trying to visualize a four-dimensional problem in Relativity, or a surface in an abstract pressure-volume-temperature space, a set of circles and squares presents much less of a challenge.

This points to a skill beyond the basic content of High School math: the ability to put various things together, and the confidence to do so.  It’s not so much dealing with a problem with many parts (though that might come into it), or remembering details about tangent and secant lines.  It is much more about using the tools one has in new situations.  We’re not yet ready to define this skill in detail or set out its characteristics, but we think it’s vital to a scientist working in a mathematical field.  And extremely useful for any problem-solver.

Unfortunately, we have no suggestions for learning this skill beyond, well, solving problems.