chocolate oranges

Every year around this time, I open up a chocolate orange and begin to wonder the same thing. As I remove wedges of the chocolate treat, is there any point at which the surface area of the partially-eaten chocolate orange is greater than the surface area of the intact orange? In other words, given a piece of foil that just covers the full orange (with no foil left over), will removing wedges render the foil insufficient to cover the remaining orange?

some sums

Had trouble falling asleep last night, so I began adding up numbers in my head. "Discovered" this interesting property of integers, though I'm sure it already exists more formally in the mathematics world:

Let n be a positive integer. Let S be the sum of all integers from 1 to n, inclusive.

If n is odd, then S is a multiple of n. However, the same property does not exist for even numbers.

It will take me some time to try to even wrap my head around this, and who knows if I even have the ability to do so. But pretty cool nonetheless. And a bit frightening that this is what goes through my head in the wee hours of the morning...

correlation?

I have noticed more white hairs on my head since getting married...

counting birthdays

happy (26th) birthday, nick!

Because math, in so many ways, in grounded in counting, I spend a lot of time thinking about how we quantify things in our lives. I've also been thinking a lot about birthdays recently, as I find myself surrounded by many friends and family members who were born in this tenth month.

What has always bothered me most is that a person can truly have only one birthday, to be celebrated on the day of one's birth. Everything that comes after represents an anniversary of one's birth day. But this is a minor point, an argument that I concede I will never win.

The argument I will win, however, is the following: We almost always incorrectly count birthdays. For example, the date on which one turns 25 is actually one's 26th birthday! But look at the Hallmark cards that arrive in the mailbox and at the writing on the cake and they will all say "Happy 25th Birthday!" Has anyone ever looked at these and said, "Hm... These are a year late."

anti-discounting

A few friends and I came across this parking meter in Amherst, MA a few weeks ago. We have not doctored this image at all.



1. What's wrong with the picture?
2a. If the meter accepted pennies, how many minutes would a penny buy you?
2b. If the meter accepted half dollars, how many minutes would 50 cents buy you?

a complete fail

I often turn to Entanglement when I'm looking to procrastinate. Unfortunately for me, I encountered this tile the other evening to start off the game:



It's as if someone knew my intentions and wanted to stop me dead in my tracks. Well, message received and understood clearly!

As for Entanglment and tiles, I now wonder how many different tiles exist in the game, and the probability of landing this tile again as the first move. Anyone know?

geometry's ironic assumption

I have emphasized over and over again to my Geometry students that "eyeballing" is never an acceptable justification for any argument. Today, I tried to make my stand a bit more compelling by presenting a series of optical illusions, a move to point out how unreliable our perceptions may be. Two lines that look congruent may actually be different lengths, or vice versa. An angle that looks like a right angle may not measure 90 degrees at all. Is that quadrilateral really a square? "Well, it looks like it is!" they tell me, exasperated with my unwavering prompts of "But do you know for sure?"



The reality is that our eyes and the sense that our brains try to impose on the world around us are completely unreliable. How we interpret the world may not be an accurate representation of the world. Or rather, what we "see" is simply that -- a representation. The images presented here are just two of a series of optical illusions I showed to my students, though these two most closely address the types of incorrect assumptions they might make with their work.

Ironically enough, students are allowed to make the assumption that a line is straight. Yet the following illusion suggests that even with straight lines, what we see is not actually truth.


Are the two red lines in the middle straight or curved?