At the close of the school year, after they complete their exams, Upper School students enrolled in William Adler’s AP Calculus BC class use the skills they’ve gained to try to solve some of the most complex problems presented throughout the history of mathematics. Working in groups of two or three, the girls choose a challenging question to tackle from the book 100 Great Problems of Elementary Mathematics by Heinrich Dorrie.
According to Adler, “the word ‘elementary’ in mathematics can be deceiving. It does not mean easy by any means. It usually means that the proof did not require calculus or set theory. What makes these problems so challenging is that they require the students to interpret and fill in the missing pieces of the proof, use technology to develop a visual lecture, and present it to faculty and peers. The proofs require the students to think strategically and experience, collaboratively, what it is like to be a mathematician.”
When it comes to the complexity of the problems the HB students took on this year, reader reviews of Dorrie’s book help to put things in perspective:
Wonderful book for anyone who A.) is a genius; B.) truly loves mathematics; and C.) doesn't mind discovering that there are things that can be done with algebra that they never dreamed of!
I love this book, and recommend it very highly if you're the type who would like to understand, say, why the Fundamental Theorem of Algebra (every polynomial equation as a (possibly complex) root), is true. Yes, it takes intellectual effort to follow the proofs, but that can be incredibly rewarding, once you finally understand.
If these descriptors pique your interest, Adler highly recommends that you check out Dorrie’s book this summer or any time you really want to put your brain to work.
In the meantime, Lauren Gillinov ’17 and Regan Brady ’17 have shared their presentation for how they solved Edouard Lucas’ Problem of the Married Couples, which asks, “In how many ways can n married couples be seated about a round table in such a way that there is always one man between two women but no man is ever seated next to his own wife?” You can find the impressive steps they took and their well-reasoned explanation here.
This problem was first presented by Lucas in 1891 in his Théorie des Nombres. When it comes to this complex question, English mathematician Rouse Ball said, “the solution is far from easy.”
Congratulations to Lauren and Regan and all of Mr. Adler’s AP Calculus BC students. We can’t wait to see what great problems you’re going to solve next.