Application

Application Geometry

CD Packing Problems

I consider myself an expert arranger of things. I enjoy rearranging storage space, packing things away, and helping people fill up moving trucks. It’s a way to apply geometry and optimization techniques, two of my favorite things.

In general, the packing problem entails trying to find the most efficient way to pack a certain kind (or kinds) of object into a certain fixed space. Packing problems are, generally speaking, very challenging because every packing problem is unique. There isn’t a good, efficient procedure that solves them all.

Here is yet another example of problems with packing problems. After shedding a bunch of CD cases, I thought I’d try to pack them up in a box. Here was my first attempt.

I got 49 CDs in the box, but there was a bit of unused space left over. I couldn’t fit a CD into that unused space, but I thought maybe I could rearrange everything to make some of that space usable.

So I tried again.

The number of CDs in this new arrangement differed by one. While I can compare which of these packings is more efficient, the problem is comparing all possible packings! There are a lot of options to consider.

As useless as they are, I ended up having a lot of fun with these CD cases. I made some parallelepipeds with them and used them to demonstrate Cavalieri’s Principle!

By patrick honner, 13 years8 years ago

Application Probability Teaching

Google and Conditional Probability

Conditional probability is one of my favorite topics to teach. Whereas normal probability calculations simply compare favorable outcomes to total outcomes, conditional probability allows us to consider the impact of certain knowledge on the likelihood of those outcomes.

For example, the probability of rolling a 6 on a six-sided die is 1/6, but if it is known that the number showing is greater than 3, then the conditional probability that a 6 is rolled is 1/3.

There are many applications of conditional probability, but a recent “Math Encounter” from the Museum of Math made me aware of an application of conditional probability that all of us see on a regular basis: Google search autocomplete.

Suppose I type in the search term “under”:

Here, Google is trying to autocomplete my search query. In essence, Google is trying to guess the next word I’m going to type. How does it make its guess? It computes a conditional probability!

Google has a lot of data on when words follow other words. When I enter “under” into the search bar, Google looks for the word/phrase with the highest conditional probability of being next. Here it turns out to be “armour”; the word with the second highest conditional probability is “world”, and so on.

Naturally, as more information is provided, the conditional probabilities change.

A fascinating, and perhaps surprising, application of a powerful mathematical idea!

By patrick honner, 13 years4 years ago

Application Resources

Free Planetarium Software

What a cool idea: a free software package that simulates a planetarium on your PC!

http://www.stellarium.org/

Preloaded with 600,000 stars, all you have to do is set your coordinates and start gazing!

You can search by name for various heavenly bodies, map the constellations, and even get close-ups of observable planets!

By patrick honner, 13 years4 years ago

Application Teaching

Testing Maximum Performance

This is an interesting article by Jonah Lehrer of the Wall Street Journal about the limits of standardized testing.

https://www.wsj.com/articles/SB10001424052748704471904576230931647955902

Lehrer discusses the results of a study from the 1980s in which psychologist Paul Sackett attempted to measure the speed of supermarket cashiers. A short “check-out test” was developed which involved scanning a small number of items. The test was administered, and resulted in a list of the fastest cashiers.

What is interesting is that when Sackett compared the results of the test with long-term data collected by the electronic scanning systems, there was a surprisingly weak correlation between the results of the speed test with the data from regular usage. That is to say, there was no real connection between being fast on the test and being a fast on a day-to-day basis.

Sackett’s misconception, and perhaps one held by many, is that there is a natural correlation between maximum performance (that on a short test) and typical performance (that is, under normal, day-to-day circumstances). Tests like the SAT, the GRE, and other high stakes tests, are tests of maximum performance. Our educational system relies on these more and more, but are we sure they measure what we assume they measure?

Lehrer points out that individual success is determined more by character traits like perseverance and self-control, but of course, it’s hard to capture that in a timed, multiple choice exam.

By patrick honner, 13 years4 years ago

Application Sports

NFL Draft Math: 2012

It’s almost time for the 2012 NFL Draft, and I thoroughly enjoy the many quantitative aspects of this event. The NFL Draft offers a complicated optimization problem with 32 actors all trying to maximize their gains.

To begin, teams and scouts evaluate the draft-eligible players and attempt to quantify their value. In doing so, they consider not just the player’s skill and athletic ability, but also the importance of the position. For example, generally speaking a left tackle is seen as providing more long-term value than, say, a cornerback.

But quantifying player value is just one part of a complicated equation. Teams need to balance player value with team need; if the best player available doesn’t fit with what the team needs, selecting that particular player may not be the best use of that pick. However, if that player is coveted by others, the team can try to extract more value by trading the pick for other picks or assets.

In order to facilitate deals, a trade value chart exists which allows teams to compare the values of different picks in the draft, almost like a currency conversion chart. It is interesting that the perceived value of picks seems to decline exponentially. And as teams package picks to move up in the draft, they may end up paying more than market value.

Further complicating matters is how player contracts play an increasing role in draft evaluation. Highly drafted players earn large guaranteed salaries, but certain positions may not be seen as worthy of such payouts. Would $10 million be better spent on an above average lineman, or an outstanding safety?

There’s a lot of math in the NFL draft, so if you like football and mathematics, sit back and enjoy! We’ve already got one great question to keep an eye on this season: will Robert Griffin III prove to be worth the high price the Redskins paid to draft him?

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