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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.]

01/26/2016 — Happy Permutation Day!

Today we celebrate the first Permutation Day of the year!  I call days like today permutation days because the digits of the day and the month can be rearranged to form the year.

01262016

Celebrate Permutation Day by mixing things up!  Try doing things in a different order today.  Just remember, for some operations, order definitely matters!

Math Photo: A 26-gon

Thanks to the strong presence of mathematicians and math teachers on social media, I know I’m not the only person who, when sitting around with a box full of compasses, unexpectedly ends up doing things like this.

Compass-gon

A circle made of circle-makers!  Or, from another perspective, a regular 26-sided polygon.

26-gon 2

The fact that I created an object that doesn’t even appear to have a name made me feel a little better about this use of time.

TCM at NCSSM

falling pebbles 3I’m excited to be presenting at the upcoming Teaching Contemporary Mathematics (TCM) conference at the North Carolina School of Science and Mathematics (NCSSM).

NCSSM is an internationally known leader in providing advanced math, science, and engineering education to public school students.  They have a residential program that serves 11th and 12th graders and distance programs that serve students all across the state of North Carolina.

NCSSM hosts the annual TCM conference to bring together teachers to talk about innovations in teaching modeling, technology, and problem-solving in advanced high school courses.  My talk, Mathematical Simulation in Scratch, details some of the work I and my students have done in our mathematical computing course.

TCM runs January 29-30 on the campus of NCSSM in Durham, North Carolina.  You can find out more information about the TCM conference and see the schedule of talks 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.