CDs, Prisms, and Parallelepipeds

I’ve had some fun playing around with old CDs and CD cases recently.

In addition to demonstrating Cavalieri’s principle both with discs and their cases, I’ve found some other mathematical uses for these objects.

Here I’ve used a stack of cases to demonstrate the difference between some geometric solids.

cd-collage

On the left, we have a right rectangular prism.  Give that prism a slight push in a direction perpendicular to a side and you get an oblique rectangular prism.

And if you give that original prism a push along a diagonal of the base, you’ll get one of my favorite geometric objects, a parallelepiped.  It may not be the perfect parallelepiped, but I’ll take it!

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NFL Draft Math: Analysis

After posting some thoughts on the mathematics of NFL Draft Pick Valuation, let’s take a look at what actually happened during the 2011 NFL Draft and see just how much credence NFL teams put into to the Trade Value Chart.

A quick re-cap:  the Trade Value Chart estimates what should expect in return for trading their pick.  The valuation of picks resembles exponential decay, and the draft format may also play a role in determining a pick’s value.

There were thirteen trades of draft picks during the first two days of the draft (Rounds 1 through 3), so I collected all of the data and tallied up the nominal values of the picks based on the chart.  Here are two examples of a trades:  the Picks row contains the picks that the team traded away; the Total Value row contains the sum of the values of the picks based on the Trade Value Chart.

trade-analysis-chart
In both cases, the larger total value is a little bit more than 10% of the smaller value.  Here is a summary of all the trades in Rounds 1 through 3.

trade-analysis-chart-summary

A couple of notes:  first, when future picks were involved in a trade, I assumed that the future pick would be the same pick number as this year’s pick, and I valued it as such.  That is, I assumed that the 2012 first round pick Atlanta sent to Cleveland would be number 27 overall, and therefore its value would be 680 points.  For percent difference, I measured the difference relative to the smaller total value.

The relative values seem close enough to suggest that the teams are using the Trade Value Chart as a guide, a conclusion also reached by @EmergentMath in his interesting analysis of past Draft day deals.

The notable exceptions in this year’s draft both involve New England, a team that is highly regarded for its skillful draft management.  They have a exceptional record for team-building, so maybe some serious mathematical analysis of draft strategy has paid dividends!

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More NFL Draft Math: Pick 33

nfl draft 2-- staffordI recently discussed some mathematics involved in the NFL Draft:  teams use a Trade Value Chart to rate the value of the different picks in the draft, and how the picks are “priced” leads to some interesting consequences, which I covered in this post.

The value of pick number 33 has fostered some interesting conversation, brought about by changes in the way the draft is run.  Originally conducted over two days with Round 2 immediately following Round 1, the NFL in 2010 reorganized the draft to take place over three days.  Now, Round 1  takes place on a Thursday night and then Round 2 begins a full day later.  (The change was presumably made to increase TV ratings, and it seems to have worked).

Now the first pick in Round 2 is the 33rd overall pick.  Noted football writer Peter King speculated last year that this new format would increase the value of pick 33, offering the following argument: with a full day to re-evaluate the remaining players after Round 1 ends, demand for the 33rd pick would grow as teams realized that coveted players were still available.  Increased demand would therefore increase the value of the pick.

The argument sounds reasonable, but ultimately I don’t think the value of this pick changes significantly.  I’ll offer both mathematical and practical counter-arguments.

First, the mathematical counter:  let’s start by noting that in any sensible rating of draft picks, the value of the picks should strictly decrease as the pick number increases.  For example, pick 21 should never be more valuable than pick 20.  Every player available at pick 21 is also available at pick 20, and at pick 20, there will be one additional player available.  In short, pick 20 gives you all the value of pick 21, plus a little more.

If the new format did increase the value of pick 33, it couldn’t raise the value above that of pick 32.  If you accept the values in the Trade Value Chart, this means the value of pick 33 (580 points) could not rise to more than the value of pick 32 (590 points), which means the most the pick could increase in value is around 1%.

Now, the practical counter:  NFL teams invest a lot of time, money, and resources into draft preparation.  It doesn’t seem likely that any team would be caught off-guard by which players are available at pick 33.  They should be just as prepared for that scenario as they are at pick 32, or pick 1, for that matter.

If a team does find itself scrambling between Round 1 and Round 2, dramatically re-valuing pick 33, then I’d that the team wasn’t properly prepared overall to begin with.  I would also suggest that team call me in for an interview!

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Math Quiz: NYT Learning Network

japan radiusThrough Math for America, I am part of an on-going collaboration with the New York Times Learning Network.  My latest contribution, a Test Yourself quiz-question, can be found here:

http://learning.blogs.nytimes.com/2011/04/27/test-yourself-math-april-27-2011/

This question is based on the increasing radius of evacuation around the Fukushima Daiichi Nuclear Plant, in the wake of the recent earthquake in Japan.


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