The optimist believes the glass is half full. The pessimist is afraid that it is.

This is the finale of the math competition scene in Mean Girls. I can only guess that references to this scene are ubiquitous in calculus classes these days (either to the delight or exasperation of calculus teachers).

Here’s a graph of the actual function, referred to in the movie as an “equation”.

I was re-watching Casino Royale recently and noticed some nice fractal imagery in the movie’s title sequence. I wonder what the Hausdorff Dimension of these fractals would be.

When the current temperature is greater than the predicted high temperature for the day, why doesn’t this report get updated? Is there a benefit in comparing the current temperature to the predicted high? Later in the day, wouldn’t it also be beneficial to know what the actual high for the day was?

One of the neat WordPress plug-ins I use is Word Stats. Prior to this post, I had apparently written over 25,000 words for this blog. That’s half way to a NaNoWriMo novel! My favorite word seems to be “that” followed by “this”. The only “math” words to crack the top 20 are “number”, “numbers”, and “triangle”.

I wonder how the Word Stats plug-in handles the mathematical notation I often use. For example, is the phrase “a + b” considered to be three words?

For example, the most widely played poker game Texas Hold ‘em allows players the possibility of using a total of seven cards to make the best five-card hand. So does seven-card stud. Omaha Hold ‘em allows a maximum of nine cards, while in Triple Draw, a player could potentially see thirteen cards before making a poker hand.

I wonder how these different games change the five-card probabilities of respective hands, and it would be interesting to see if it changes how any of the hands stack up in relation to each other.

But as it turns out, this is so common among the real numbers that given any two real numbers x and y, there is a third number z such that x + y + z = xyz as long as xy ≠ 1. This is fairly easy to show with some algebra.

What does a plot of all these points (x, y, z) look like? Let’s turn to WolframAlpha to give us a 3d graph:

I expected the above to be a little more interesting until I realized the boundaries were too narrow. Here’s a better look:

Interesting!

How many balloons are these two men carrying? Questions like this just come to mind when I’m walking down the street.

If I could have a mundane superpower — a knack or skill that is extremely useful but much less impressive than flying, invisibility, or super strength — I would choose to be an amazing estimator. If I could accurately estimate any quantity or measurement, I think I could do a lot of good.

I recently learned that the Chinese calendar has a different definition for a leap year. The Chinese calendar is primarily a lunar calendar. That means that months are determined by the phases of the moon. Specifically, the start of each month always occurs on a new moon. A typical year has twelve months and 353, 354, or 355 days. The correction to account for the difference between a Chinese calendar year and an astronomical year is a little more drastic: every so often, a new month is inserted because there will be thirteen new moons within a year.

Complicating matters even further is a rule that states the winter solstice must occur in the 11th month. Therefore, the point in the calendar where the new month is added will vary from leap year to leap year. How often do leap years occur in the Chinese calendar? The intervals are not regular, but it’s once every three years!