I recently heard a first-grade teacher espouse the benefits of having students create equations that are both true and false. Her argument was that students typically only see equations that are true, but by considering equations that are false, students develop a better understanding of equality and the equals sign.
She showed us a stack of equations, both true and untrue, that her students had created. One of them read:
6 – 6 = 9 – 9
This equation happens to be true, but what struck me was its symmetry. I commented, “That one is especially nice, since it’s also true when you display it upside down.” Of course, this is a trivial example of an equation that remains true when rotated 180 °, but it led me to consider a bigger challenge:
Can you find a non-trivial example of an equation that remains true when rotated 180°?
What is the most complex example of an equation that remains true when rotated 180°?
I think today (9/6) is a good day to share this puzzle with you, since today’s date, minus the year, remains the same when rotated.
I was able to discover a few (fairly basic) answers to the first question, but I won’t share them here for fear of spoiling your fun. As for the second question, I don’t have an answer; in fact, I’m not even sure it’s answerable.
Here’s a poem to remember the procedure for dividing fractions, which happens to reference inversion:
Ours is not to reason why;
Just invert and multiply!
An inversion is a word or phrase that reads in more than one way. Most often, an inversion involves writing letters so that they have rotational or reflexive symmetry — that is, the word or phrase reads the same when turned upside down or when viewed in a mirror.
I’m a rank amateur, and my best inversion is shown below.
Scott Kim created an inversion of my name, too, and his version is shown below.
I think mine is better, but there are several major difference between the two inversions:
- I created mine on a computer, so it has perfect rotational symmetry and nice clean lines. Scott Kim drew his freehand.
- My version is only my first name. Scott’s version contains both my first and last names.
- The inversion that I created took several weeks and many refinements to complete. The inversion created by Scott Kim took him less than 30 seconds. (No, seriously — he did it on the spot after I spelled my last name for him.)
Finally, let me tell you the story about the first time I met Scott Kim. I used to work for the MATHCOUNTS Foundation, and one of my responsibilities was finding a speaker who could entertain 228 middle school students each year at our national competition. In 1996, I contacted Scott, and he agreed to give a talk. I was ecstatic that I was able to convince him to speak to our participants, but at our weekly staff meetings leading up to the national competition, my boss would continually question my choice. “Who is he?” Camy would ask. “What does he do?” At five consecutive staff meetings, I tried to explain what it is that Scott Kim does. I even showed her examples, but she just never got it. Finally, at the sixth staff meeting, the questions came again, and I finally responded, “I’m sorry that I’m not able to get you to understand how cool he is. But you’ve got to trust me, Camy — the kids are going to love him.”
When Scott Kim arrived at the national competition, I was unfortunately busy, so I quickly introduced him to my boss, and then I excused myself. Consequently, I wasn’t present for the rest of the story, but this is how it was relayed to me.
“Camy,” said Scott Kim. “That’s an interesting name. I don’t think I’ve heard it before. Is that with one M or two?”
“Just one,” Camy said.
Scott thought for a moment, and then scribbled something on his notepad. He ripped off the sheet and handed it to her.
She looked at it. “Yes,” she said. “C-A-M-Y.” She was clearly unaware of what she was looking at.
Scott said, “Actually, it’s not just your name. It’s an inversion.” He then turned the paper upside down for her, and there was her name again.
“Oh, my God!” she screamed. “That is so cool!”
Ha-rumph. In less than five seconds, Scott Kim was able to do what I couldn’t do in five consecutive staff meetings.
But it gets better. After chatting for a few minutes, Camy introduced Scott Kim to another celebrity who was attending the competition — Scott Flansburg, who is better known as The Human Calculator. Scott Flansburg had just witnessed what Scott Kim had done with Camy’s name, so when Scott Kim scribbled something on a piece of paper and handed it to him, he looked at it and said, “Oh, I get it. It says Scott Flansburg, and when I turn it upside down, it’ll say Scott Flansburg again.”
“No,” said Scott Kim. He then turned the paper upside down for Scott Flansburg, who read aloud what was now showing on the paper: “The Human Calculator.”
That’s right. In less than 20 seconds, Scott Kim created an inversion that read “Scott Flansburg” in one direction and read “The Human Calculator” when rotated 180°.