Fun with e

Found a fun problem via facebook last week: find a set of whole numbers that add up to 25, and maximize the product of these numbers. It’s such a fun little puzzle that I won’t give away the solution here. But I got interested in another issue that my friend Aaron pointed out: the connection to e as the “best base”. To get some idea about what I’m getting at, let’s stay with 25 and find the set of numbers (not necessarily whole numbers) that add up to 25 and maximize the product. Actually the numbers have to all be the same — if any two numbers were different, then you could move them closer together and thereby increase their product. So basically what we’re looking at is to maximize the expression (\frac{25}{n})^n for positive integer n.

I hope you’ll play around with this REALLY COOL GEOGEBRA APPLET (screenshot below) before continuing.

geogebraScreenshot

The thing that really interested me in this problem was how changing the value of A (originally 25) would change the optimal value of n. Sure, for 25 we see that the best value is n=9, but apparently for some larger value of A, then we need n=10. What is the cutoff? When do we switch from 9 to 10?

In fact, you can try it: increase A gradually until the line for n=9 is as high as the line for n=10. You should find yourself at A=25.81 or so. To find the exact value we can do a little algebra. Suppose we have A such that

(\frac{A}{9})^9 = (\frac{A}{10})^{10} .

We can do some rearranging…

A=\frac{10^{10}}{9^9}

Et voilà! A similar calculation will show that when A=\frac{9^9}{8^8} , both 8 and 9 groups give the same product.

Another issue implied by the applet is the link to e. Apparently, we would like the ratio A/n to be close to e to get the maximal product. It’s no coincidence that our original value of 25 is close to 9e (about 24.46). This leads to the interesting inequality

\frac{9^9}{8^8} < 9e < \frac{10^{10}}{9^9}.

And based on the context, this is the range where we would choose to divide A by 9, and 9e should be more or less in the middle of this range.

9e

Of course, there’s no reason not to extend this a little on both sides…

\ldots < 7e < \frac{8^8}{7^7} < 8e < \frac{9^9}{8^8} < 9e < \frac{10^{10}}{9^9} < 10e < \frac{11^{11}}{10^{10}} < 11e < \ldots

7e11e

Just go ahead and generalize that…

\ldots < (n-1)e < \frac{n^n}{(n-1)^{(n-1)}} < ne < \frac{(n+1)^{(n+1)}}{n^n} < (n+1)e < \ldots

Turn your eyes to that last piece on the right.

\frac{(n+1)^{(n+1)}}{n^n} < (n+1)e

We can simplify a little…

\frac{(n+1)^n}{n^n} < e

(\frac{n+1}{n})^n < e

(1+\frac{1}{n})^n < e

Where have I seen that before?

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Mystery of the Missing Hummus

IMG_2494

Mmmmm…. edamame hummus, two delicious flavors in one perfect cylindrical container. 8 ounces of goodness.

IMG_2496

Well this is alarming. Couldn’t they fill it all the way? How much hummus am I missing?

IMG_2495

Wait, that’s cheating!

Time to put on your geometric modeling caps! What questions would you ask? What do you need to know?

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meta-post / about writing

The dreaded meta-post: posting about posting. Generally I am able to resist the temptation, but now I’ll indulge myself in writing about how this is my first post in three months. While I have some blockbuster blog updates kicking around in the back of the brain — just smile and nod — here I’ll be writing about why I’m not writing.

Thesis: I lack internal motivation.

Some nuance: Even though I can think of myriad moments of internal motivation, at a basic level I am waiting for external forces to spur me into action.

  • Evidence 1: My last three blog posts are due to participation in the new blogger initiative.
  • Evidence 2: Last year when I took that class about strategies for ELLs, my teaching became stronger as a result of being forced to use these strategies in my classroom. Now, I’ve mostly let this habit fade away.
  • Evidence 3: On a large scale, my best professional work in (let’s say) the past five years has been a result of the pressure of developing curriculum and plans to use in my class.

Of course (is obvious nuance possible?) most folks who aren’t freelancers and/or vagrants might write something like item 3. This is the nature of a professional career, and as humans we use each other to accomplish more than we would individually.

But to bring a merciful end to an indulgent post, I’ll relate a story my friend told to a group of aspiring writers (I went incognito). It was some question about, what is the most important lesson you learned in the process of getting published? Response (and excuse my poor sense for reconstructing remembered dialogue):

I think the big thing is, no excuses. I had this idea about writing a story about LA, and I thought that I should move to LA to write the book. A friend said, ‘why do you have to move? Just write the book here.’ I was like, … oh yeah! I don’t need to move to LA to write the book. I don’t need to wait for x to be able to do the work — just do it now. If you want to write — well, start writing.

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Which Spawned the Title, “Brain Open Now”

Not sure if I’ve ever explained the name of this here blog. “Brain Open Now”, you can see it right up there. So, what does it mean? I’ll tell ya. Let me take you back to a fellow you mighta heard about… fellow named Paul Erdős.

Click for source: http://www.facets.org/edu/n-is-a-number-a-portrait-of-paul-erdos/

Open Up!

Interesting guy, and I can point you to the obvious places to to read about him. Seems like he would come into a classroom to hear or give a lecture and declare, “My brain is open.” A rather pedestrian virtue of being open-minded is filtered through the Hungarian (and whatever else) to become this somehow medical/mechanical statement about his inner workings. Say this, and you say that you are willing to entertain and give energy to any line of thought. And to me, this also has connotations about being transparent with thought, like in the phrase “open source”, as if one could observe how the thought is formed, developed, and executed.

Click for source, for what it's worth: http://images.elephantjournal.com/wp-content/uploads/2011/08/4-dr-frankenstein.jpg

This is as good a premise as any for starting a blog!

This sounds suspiciously like blogging.

So I thought to title my blog “My Brain is Open” and get on with that. But then I found that someone on WordPress had already done so, so I struck out for (relatively) more original ground.

For some reason I chose “Open is My Brain” “Brain Open Now”, which could be part of the phrase, “My brain is open now”. But I prefer to read it as a command: “Brain! Hey brain! You there? Ok …. Open Now!”

Click for image source, http://1.bp.blogspot.com/_psW9LnYSTFU/TBg_bqvs3BI/AAAAAAAAAJg/1iEgtKsa4uE/s1600/frankenstein_lab.jpg

C’mon… I said open up!

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