I began with two assumptions.  They are debatable, like most assumptions are, but they seem like a good place to start an investigation:

1)  Every amount of change is equally likely to be received.

2)  Every amount of change is provided using the minimum number of coins.

What (1) means is that you are just as likely to get 13 cents back in change as you are to get 91 cents when you purchase something.   And (2) means that, when you get that 91 cents back, you’ll get it as 3 quarters, 1 dime, 1 nickel, and 1 penny; not 4 dimes, 9 nickels, and 6 pennies.

I made a chart in Excel of all the possible change amounts from 1 to 99.  I then figured out how many of each coin would be used to provide that amount of change, assuming that change was given efficiently.

Now, assuming each change amount is equally likely, we can simply count the total number of coins and then figure out each percentage as a share of that total.  The total number of coins in the list is 466.  The number of each coin, and it’s approximate percentage, is given below.

By this analysis,  a large, random collection of coins should be roughly 42.5% pennies, 8.5% nickels, 17% dimes, and 32% quarters.   Do me a favor:  the next time you find yourself sitting on a big pile of change, see how it stacks up against these numbers and let me know.

And if you like, you can check this theoretical ratios against the actual numbers in my Piggy Bank.

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Expressing today’s date in the typical European style [ Day / Month / Year ] gives us a palindromic date!  Definitely a cause for mathematical celebration.  Or at least mathematical appreciation.

I admit that, as an American, this doesn’t resonate with me as much as another recent Palindrome Day, but it’s always nice to take a moment and appreciate a good number.

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Inspired by Escher’s Planaria, Professor Hart designed and manufactured a linkage-system that can be used to build a face-centered cubic lattice.  After assembling the pieces, the participants in the workshop were encouraged to build away!

It’s probably only natural that we became consumed with building something tall.  And as we did, conversations arose about the role of symmetry in nature, as stability and strength in our structure seemed to demand it.

A fun, engaging, and mind-opening exploration of geometry!  And much more.  You can see more photos from this workshop on my facebook page and Professor Hart’s website.

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http://www.thatquiz.org/tq/practice.html?idpoints-j8-l5-m2kc0-na-p0

After you’ve warmed up, be sure to try plotting complex numbers, or test your knowledge of the Polar plane (in both degrees and radians).

There are a number of simple little quizzes to enjoy at thatquiz.org; just be careful not to let the day slip away!

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http://goo.gl/7YLu

The band Radiohead famously offered their album “In Rainbows” on their website and asked fans to pay whatever they wanted for the download.  The actual sales numbers are well-guarded, but  it appears to have been a success.

The above article details how an amusement park merged the pay-what-you-want approach with a half-goes-to-charity approach (telling the customer that half of the purchase price is donated to charity).  The product in question was a picture of the customer riding a roller coaster.  Let’s abbreviate with PWYW (Pay What You Wish), HGTC (Half Goes to Charity), and PWWTY (pay-what-we-tell-you):

When given the opportunity to pay whatever they wanted, participation increased dramatically, but revenue was still low–only 92 cents per person.  But when combined with the half-goes-to-charity approach, participation was much higher and the price paid was significantly higher.  Even after taking out the half for charity, revenue was still up by a factor of three!

This is a very interesting approach to pricing, and there are some cool psychological and sociological principles at work here.  And it’s another set of factors to consider when that salesperson is working on you.

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Using Google’s new graphing feature, all you have to do is plug this equation into Google’s search bar.

For

Or you can just click this link.

Start working on your Easter Eggs!

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There were plenty available, so I started tiling the plane.

Unfortunately, I didn’t have time to finish.

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We enjoyed our first permutation day of 2012 just 21 days ago.  Exactly three weeks apart!  That seems unusual.  I also wonder “How close could two permutation days be?”

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

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http://goo.gl/cHblY

Seen at right is the radial engine.  The constant velocity joint is my favorite, but it’s also great to learn how a sewing machine really works!

Some great visualizations of interesting and intricate 3D geometry and engineering.

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http://goo.gl/PkzZa

The speakers are employers and recent graduates working in technical fields, and they offer a lot of clear, thoughtful advice.

The “Why are ‘Soft Skills’ Important?” segment is particularly great to see:  the speakers all stress how communcation, writing, presenting, and teamwork skills are crucial to success in technical fields.  In addition, the “Why are Maths Graduates Sought After?” segment really highlights what is fundamentally important in mathematics:  problem solving skills, thinking creatively, and the ability to break complex ideas down into simpler ones.  These skills are prized in every field and industry.

The best advice I saw throughout the videos is to start thinking early about what really interests you and explore all the options you can.  There’s more to math than just teaching and banking!

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http://www.geometrygames.org/TorusGames/

A torus is a geometric surface that is basically equivalent to a donut (the surface of the donut; not the interior).  One way to [theoretically] construct a torus is to take a flat piece of paper, roll it into a cylinder by gluing the two opposite edges together, and then curling the cylinder into circle and gluing the two circular ends of the cylinder together.

So how do you play pool on a torus?  As the image to the right suggests, if you shoot the ball toward the top edge of the table, it will wrap around and come out the bottom.  The sides similarly wrap around, too.

This strange geometry definitely creates new strategic options in Chess and Tic-Tac-Toe, and this free set of games also include Word Search and Crossword.

A simple, thought-provoking idea.  And free and fun to boot!

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Ever since I switched from buying unlimited cards every month to purchasing cash cards as needed, I’ve been having serious problems.  I really don’t plan ahead, so I’m usually in a rush when I discover that my current card has run out.  When this happens, I’ll just quickly buy a new card with $50 on it, without thinking through the quantitative consequences.

A single ride costs $2.25.  In addition, there is a Bonus system in place where you get additional credit for putting money on your card.  It’s obviously too complicated for me to work out, because I keep ending up with cards that have less than $2 on them.

There are three of them in my wallet right now.

The value of my current card actually ends in $0.10.  I honestly have no idea how that happened.

I guess it’s time for me to review my own post about maximizing Metrocard management.

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http://goo.gl/GTdCb

A team of complex systems analysts created a map of how 40,000 transnational companies are connected to each other.  Using quantitative factors (like amount of shares of company X held by company Y, and the economic relative strength of company X), the team developed a way to measure the influence a company exerts over another.

In addition to looking at individual companies, this measurement for the evaluation of the system as a whole in terms of equity and stability.  The team found that a sub-network of 147 tightly knit companies exerted a disproportionate influence over the entire network.  The high level of interconnectedness of this group concentrated power in the network, allowing this small group of less than 1% of the companies to effectively control 40% of the network.

An innovative application of mathematics to the fields of economics, politics, and social justice!

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I thought a simple place to start examining this question would be to look at Superbowl scoring versus regular season scoring.  Below is a chart showing the difference (Superbowl Score – Average Regular Season Score) for all 46 Superbowls.

At the far right, we see the results of Superbowl 46:  Giants 21, Patriots 17.  The league average in scoring this years was 22 points per game, so the difference here is 38 – 44 = -6.

It seems as though it is more common for more points to be scored in the Superbowl than in an average regular season game.  Unfortunately, there are a lot of stories one could tell about why that might be so:  better teams (and therefore better offenses) make it to the Superbowl; defenses are more susceptible to pressures of the big game; the extra preparation time gives offensive coordinators and advantage.

So how could we more rigorously explore the quantitative characteristics of the Superbowl?

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http://video.google.com/videoplay?docid=8269328330690408516

Although the mathematics of the proof could not possibly be explained to the layperson (there aren’t many people in the world who could really understand it in its entirety), this BBC documentary does a great job of narrating the struggles, setbacks, and triumphs of Wiles’ pursuit.

The story of the hero and the many peripheral characters (including John Conway) opens a wonderful window into the world of advanced mathematics.

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http://www.metanamorph.com/

The website is a little clumsy, but there are some really cool projects here, including The Crevasse, The Waterfall, and Lava Burst.  There are also several videos showing how the artist puts these perplexing pavement paintings together.

For an interesting application of this idea, check this out:  a 3D pavement painting of a child playing in the middle of the street!  The idea was to get the attention of speeding drivers.  I bet it did just that!

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Here’s a short summary:  the first possession in overtime is determined by a coin flip.  The first-possessing team will be known as Team A, and the other team will be Team B.  If Team A scores a touchdown, they win immediately; if Team A scores a FG, then Team B will receive the ball and can win (with a TD), tie (with a FG), or lose (with a turnover).  Aside from the specific first- and second-possession situations governed by these rules, the rest of overtime plays out in classic sudden-death style.

I looked closely at whether it is useful for Team A to attempt a FG, but the question that occurred to me recently was “If you win the coin toss, is it obvious that you want the ball first?”

Consider the following chart.  Again, Team A is the first-possessing team, and Team B is the other team.  The chart shows all the possible sequences of outcomes on the first two possessions:  touchdown (TD), field goal (FG), or Turnover.  Here, a turnover could be a fumble, and interception, a punt, or a turnover on downs.

The first-possessing team, Team A, certainly has an advantage in that they can end the game immediately by scoring a touchdown.  But if they don’t score a touchdown, Team B can always win with a TD and in some cases can win with only a FG!  Furthermore, as the original kickoff team, Team B may enjoy a strategic advantage in field position, which can further affect the above scenarios.

Now, in all situations where the game is not decided within the first two-possessions, Team A gets the ball first as sudden-death rules take over.  However, I recall analyses that suggest that the first possessing team doesn’t have a statistically significant advantage in sudden-death overtime.

This is an admittedly elementary look at the situation, but it seems to me that the answer to the question “Do we want the ball first in OT?” is not obvious.  Hopefully I’ve given Coach Belichick and Coach Coughlin something to think about this week!

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http://goo.gl/84bt1

This question is based on the late Joe Paterno’s record for major college football victories by a head coach.  How many years would it take for a contemporary to catch him?

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The same kinds of issues are generally present.  There are instances of mathematical errors, poorly constructed questions, underrepresented topics, and 9th-grade questions on 11th-grade exams.  Here is a quick overview of the Algebra 2 / Trigonometry Regents, the highest-level state math exam in New York.

Mathematical Errors / Poorly Constructed Questions

The exam writers for the New York Regents exams continue to find new and innovative ways to construct erroneous questions.  Here is number 23 from the multiple choice section:

Which calculator output shows the strongest linear relationship between x and y?

This is a bad question to begin with, in that it really isn’t a math question.  The student isn’t being asked to ponder anything mathematical; instead, the student must recognize a step in an artificial procedure about which they have no real mathematical understanding.  (The mathematical technique for finding linear regression equations is not taught in this course; it is expected that the student will use the calculator to generate the equation).

What’s remarkable here lies in the “answer”.  The way you assess the relative strengths of regression equations is by comparing their correlation coefficients (the r values).  Generally speaking, the closer r is to 1 or -1, the stronger the correlation.  In answer choice (4), the value of r is closer to -1 than, say, the value of r is to 1 in answer choice (1).  Thus, (4) is the correct answer.  Pretty easy, right?

Amazingly, the situation represented in answer choice (4) is a logical impossibility.   Since the r value is negative, this means the correlation between the two sets of data is negative; but the regression line’s slope (the b value) is positive!  This cannot happen.

As a result, a scoring correction was issued (after the exams had most likely been graded) and all students were to be given credit for this problem regardless of what answer they put.  And so another flawed question makes it through the draft-revision-publish cycle and into the hands of thousands of students.

Underrepresented Topics

By my count, only one question on this exam (worth 2 out of 87 total points) required the use of either the Law of Sines or the Law of Cosines.  Now, I don’t know how many points should be allocated for these particular techniques, but they are fundamental ideas in trigonometry:  they should be a non-trivial part of the course.

Indeed, a quick look at the exam’s reference sheet is illuminating:  less than half the formulas that are traditionally provided for the student relate to questions on this particular test.

9th Grade Questions on 11th Grade Exams

The final question, and the highest-valued question (6 points) on this Algebra 2 / Trigonometry exam, asked the student to simplify the following expression:

Once again we see a problem from the 9th-grade Algebra curriculum playing a significant role in the 11th-grade Math Regents exam.  A quick look at the official Integrated Algebra Pacing Guide shows that dividing and simplifying rational expressions, the techniques that this problem requires, are part of the 9th-grade course.

This isn’t necessarily an easy problem, and it does require dealing with a cubic polynomial (although factoring by grouping is an optional part of the 9th-grade curriculum).  But this is yet another example in the evolution of this exam showing that, year after year, the hardest questions seem to get easier.

To be completely fair, it was nice to see the exam writers include asymptotes on their graphs this time (to avoid fake asymptotes like these), and they did demonstrate a little more understanding about 1-1 functions (perhaps they did a little studying after this disaster).  But overall, it seems to be business as usual.

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http://commons.bcit.ca/math/examples/

Interested in how Linear Algebra can be applied to Nuclear Medicine?  Or how Logs and Exponentials can be applied to Forestry and Wildlife?  Or how Differential Equations can be applied to Mechanical Engineering?

Well, look no further!

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http://www.youtube.com/watch?v=JX3VmDgiFnY

This short movie visually explains all the basic transformations of the plane:  translation, rotation, dilation, and inversion.  Then it demonstrates how all of these transformations of the plane can really be thought of as simple translations and rotations of the sphere!

If I had seen this video a few decades ago, I might not have given up on topology as quickly as I did.

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http://goo.gl/Kqc86

The ideas cover both primary and presidential elections, and involve polling, demographics, campaign finance, and even a little graph theory!

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http://goo.gl/txnYF

This question is based on the rising number of weight-loss surgeries performed each year in the U.S.  How many are being performed on patients under 21 years of age?

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http://news.sciencemag.org/sciencenow/2011/03/paper-plastic-or-steel.html

Although you probably wouldn’t want to carry these to the store with you, this result could have real applications in industrial packaging.  In addition, it’s another step towards the mathematical-origamist’s dream:  designing building that can rearrange and rebuild itself as needed!

A great talk by Erik Demaine on mathematical origami opened my eyes have this amazing application of mathematics, and my students and I have been having a lot of fun with folding ever since!

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http://goo.gl/zOG4n

This question is related to a publicly-funded program that trains private-sector employees in North Carolina.  Just how much does this program cost the government?

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We enjoyed several Permutation Days in 2011.  In addition to predicting future dates, an interesting question would be “Which year has the most Permutation Days?”

In honor of today, I recommend changing the order in which you do things!

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And check out these water parabolas at the Detroit Airport!

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