What Is Mathematics?

Mathematics is creative.

Unfortunately, this is lost upon many — if not most — students of mathematics, in large part because their teachers may not understand mathematical creativity, either.  One way to address this issue is to have students write and solve their own original mathematics problems.  This seems daunting at first, until students realize they are more creative than they were led to believe.  (I’ll discuss this more in a later post.)

The difficulty is that the creative dimension of mathematics is a bit elusive.  Give a child crayons and ask her to draw a picture, sure — but give a student some ideas and ask him to create a new one?  To appreciate mathematical creativity, you need some understanding of the abstract nature of mathematics itself.  To create mathematics, you need imagination much like you do in any of the arts — or other sciences, for that matter.

Over the years, I’ve created my fair share of mathematics.  How much of it is really new is hard to determine — how do you know if any of the billions of other people in the world already created something you did?  (Proof by internet search notwithstanding.)

This blog is about sharing some of my ideas, problems, and puzzles.  Some were created years ago, some are new — and I will consider myself lucky if some are entirely original.  I truly did have fun creating them, and I enjoy writing about them now.

I’m hoping to convey an enthusiasm for mathematics and its related fields — in other words, all human knowledge — and to share something of the creative process as well.  The creation of mathematics is not a mystical process, and needs no explanation to a mathematician.  But we can surely do more to make this enlivening process accessible to all in a time when it is certainly necessary.

As you follow, you’ll notice a heavy emphasis on programming.  Every student should learn to program — and in more than one language.  Perhaps this should be an axiom in the 21^st century, but we’re not even close.  So many of the tools I use are virtual — the ability to write code to perform various tasks is essential to my creative process, as you’ll see.  In fact, many posts will have links to Python programs in the Sage platform (don’t worry if you don’t know what these are yet).  These tools are all open source, and available to anyone with internet access.

Finally, blog posts will usually have a “Continue reading…” section.  Some posts (like this one) will be essays on teaching, creativity, or a related topic.  Since not everyone may be so philosophically minded, the “Continue reading…” sections of these essays will be a puzzle or game.  Enjoy!

You’ll find out that I usually don’t give answers to the problems — rather hints.  The hint for this puzzle is that the total cost of all the ice cream treats is $17.25.  It’s pretty unlikely you’d end up with the correct total if you didn’t solve the problem correctly.  But with the solution given, it may be too tempting to peek.  Rather a metaphor for life — no Solutions Manual provided.  Now onto the puzzle!

Al, Bert, Chuck, and Dee went to Ed’s I-Scream Emporium to get some ice cream!  Each got a different delectable concoction.  From the following clues, can you figure out what each ordered?

1. A cone costs $2.50 for one scoop, plus $1.15 for each additional scoop.  A cup costs $2.00 for one scoop, with $1.15 for each additional scoop.  Toppings are $0.35 each.  A Brownie Sundae costs $4.75 and has three scoops of ice cream.  You can’t get toppings on a Brownie Sundae.
2. Al got the same number of scoops as Dee got toppings.
3. Bert and Chuck got the same number of scoops.
4. Only one of the four had the same number of scoops as toppings.
5. Only one of the four ordered a Brownie Sundae.
6. Dee got the same number of toppings as the numbers of Al’s and Bert’s toppings combined.  (None of these numbers is 0.)
7. The number of Bert’s toppings was the same number as the number of cones ordered.
8. A cone can only have one topping.
9. Anyone who didn’t order a sundae got at least one topping.
10. Bert did not order a cone.

If you’re interested in a printer-friendly version of this puzzle for personal or classroom use, you may download it here: Day001Puzzle