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Archive of posts filed under the Application category.

SIAM ED16

siam-ed16I’m excited to be heading to Philadelphia this weekend for the SIAM Conference on Applied Mathematics Education (SIAM ED16).

I’ll be presenting on the work I do with mathematical simulation in Scratch, and I’m really looking forward to the variety of talks on bringing applied mathematics and computing into classrooms.  In particular, I’m excited to hear Maria Hernandez from NCSSM talk about how to teach modeling and Gil Strang from MIT talk about the teaching of Linear Algebra.

You can learn more at the conference website, and see the full conference schedule here.

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Packing Trapezoids

I find interesting applications of mathematics every time I visit IKEA.

Packing Trapezoids

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Superbowl Predictions

ESPN recently published a list of “expert” predictions for Superbowl 50.  Seventy writers, analysts, and pundits predicted the final score of the upcoming game between the Carolina Panthers and the Denver Broncos.  I thought it might be fun to crowdsource a single prediction from this group of experts.

Below is a histogram showing the predicted difference between Carolina’s score and Denver’s score.  The distribution looks fairly normal (symmetric and unimodal).

Super Bowl 50 Predictions

The average difference is 6.15 points, with a standard deviation of 7.1 points.  Since we are looking at Carolina’s score – Denver’s score, these predictors clearly favor Carolina to win, by nearly a touchdown.

This second histogram shows the predicted total points scored in the game.  The average is 44 points, with a standard deviation of 5.7 points.

Super Bowl 50 Predictions -- Total Points

Combining the two statistics, let’s say that the group of ESPN experts predict a final score of Carolina 25 – Denver 19.  We’ll find out just how good their predictions are tomorrow!

[See the full list of ESPN expert predictions here.]

How Much Would You Pay for a 20% Discount?

A local Office Max is going out of business and is having a very interesting sale.

20 percent discount 2

I’m not sure I’ve ever seen a sale where you earn a discount by purchasing a certain number of items.  Of course, I immediately began exploring the mathematical consequences of the policy.

The first thing that occurred to me was that you can essentially purchase a 20% discount.  Say you need to buy n items.  Simply buying another 20 – n items earns you a 20% discount.  The natural question is thus, “Under what circumstances would buying an additional 20 – n items be worth a 20% discount?”

There are a variety of factors to consider.  For example, if you can just find an additional 20 – n items that you are happy to buy, it’s definitely worth it:  you get the 20% discount, and you get items of value to you.  Also, the answer likely depends on n:  if you are only 1 item short of the discount, it’s easier to justify an unnecessary purchase than if you are, say, 19 items short.

As an extreme case thinker, I considered the following scenario.  Suppose I wanted to buy one item; under what circumstances would I buy 19 items I didn’t want in order to get a 20% discount?

Obviously, the key to this strategy is finding a cheap item to purchase 19 times.  I thought I had found the cheapest possible item here:

composition books 74 cents

Nineteen composition books would cost me $14.06.  If the 20% discount saved me more than $14.06, this strategy would be worth it.  This sets the bar for my one item at $70.30.

However, I later realized I could do better here:

paper folers 89 cents

These paper folders cost more per item, but unlike the composition books above, the folders are themselves eligible for the 20% discount!  Nineteen folders would cost $16.91, but they’ll be discounted 20% to $13.53.  This means if my single item cost more than $67.65, this strategy would save me money.

I could have done a lot better if these Slim Jims were sold here, or these 10-cent envelopes!  But this is the best I could find in the store.

Another interesting question to consider is “For what range of prices would buying nine additional items, to receive a 10% discount, be a better strategy than buying 19 additional items, to get the 20% discount?”

In any event, I appreciate Office Max giving me something interesting to think about as I waited in line.  And as usual, I waited a very long time.  Let’s just say it’s no surprise they are going out of business.

Hands on a Subway Pole

When I describe the role mathematics plays in my life, I often say that it gives me a set of tools to process and understand the world.  One way that manifests itself is that I see graphs everywhere.

For example, when I look at a pole on the subway, I see the distribution of hands that have been on the pole.

hands on a subway combo

I think about things like this because they are interesting, but also because they are practical.  Where is the pole the dirtiest, and cleanest?  Where are germs most likely to reside?  New Yorkers know instinctively to touch as little as possible, but sometimes you have no choice.  Best to know your probabilities ahead of time.