Category Archives: Problem Solving

Ancient War Tricks and Mathematics

Quanta Magazine has an interesting article about how an ancient Chinese warfare technique is used in mathematics today, including making predictions for…comets? Check out the article here.

BADGING:

Read the article linked above.

Then, answer the following questions.

  1. One of the first scenarios in the article states: “In a morning drill you ask your soldiers to line up in rows of five. You note that you end up with three soldiers in the last row.” Make a list of the first 10 possible number of soldiers that could fit this arrangement.
  2. One of the next scenarios in the article states: “...then you have them re-form in rows of eight, which leaves seven in the last row.” Make a list of the first 10 possible number of soldiers that could fit this arrangement.
  3. The third scenario in the article states: “…and then rows of nine (soldiers), which leaves two (soldiers in the last row).” Make a list of the first 10 possible number of soldiers that could fit this arrangement.
  4. Explain how, once you found the first number shared by each of the three lists above, you could “jump ahead” to the next number shared by each of the three lists.
  5. Define the term “pairwise coprime“. Give an example of two numbers that are pairwise coprime.

The Perfect March Madness Bracket

If you’ve ever filled out a March Madness bracket for the NCAA tournament, you probably realize that you aren’t super likely to pick all of the games perfectly. See how bad your chances are here.

BADGING:

Read the article linked above. Answer the following questions in a couple of sentences each.

  1. How do we arrive at the odds of 1-in -9.2 quintillion? Explain the mathematics that gives that large of an answer.
  2. One mathematician actually estimates that your odds are between 1-in-10 billion and 1-in-40 billion. Explain why he feels that you have these slightly better odds than the original 1-in-9.2 quintillion.
  3. Using the national accuracy of 66.7% when picking first-round games, and assuming we could continue that success rate over the course of the entire tournament, what are the odds of picking a perfect bracket?
  4. Over the past 8 years of bracket challenges, winners have averaged picking 49.8% of games correctly. Using that percentage and the techniques you saw used in the first three questions, what are the odds of a winner predicting a perfect bracket?

Thanks to Hugo W. for suggesting this badge!

How Half-a-Million Home PC’s Finally Cracked an “Unsolvable” Math Problem


Many people’s home computers sit idly during the day when homeowners are away at work or school.  Did you know that some organizations allow you to connect your computer to a mainframe so that they can “borrow” bits of your operating power to work on difficult problems?  The Charity Engine is one, and it helped to solve one of history’s great unsolved math problems.

BADGING:

Watch the the first 5 minutes of the Numberphile video embedded above, and then read this brief Popular Mechanics article.

Answer the following questions in a few sentences each:

  1.  Describe the “sum of three cubes” problem (aka a “Diophantine equation”).
  2.  Explain why some numbers (like 4 or 5) will never be written as a sum of three cubes.  What mathematical property do these numbers share that makes them unwritable in this way?
  3. Why are 33 and 42 “special cases” when it comes to Diophantine equations?
  4. Explain how long it took computers to finally find a Diophantine solution to 33 and 42.
  5. Find any two Diophantine solutions/equations that weren’t shared in the video or the article.

Thanks to Mr. Victoria for sharing this article as a badging opportunity!

Number Gossip

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“Can you believe what 56 did?  It’s just so…odious!”

“Oh I know.  And 43 is so lucky, I can’t even stand it.”

You probably know a lot of properties of numbers like “even”, “odd”, “prime”, “square”…but there are so many more that you might have never heard of!  Head on over to Number Gossip to get the scoop!

BADGING:

Pick a favorite or interesting whole number.  It might be your uniform/jersey number for a sport you play, or your home address, or your lucky number, or something else entirely.  Enter it into the search field at Number Gossip.

  1.  List all of the “common properties” of your number that Number Gossip lists.  If any of those properties are unfamiliar to you, you should be able to click for an explanation.   Explain in a sentence next to each property why your number belongs to that property (Where applicable, give a specific reason for *your* number, not just a definition of the property).
  2. Pick one “rare property” (if your number has one; not all do) and do the same thing as in step 1.
  3. Pick one “unique property” (if your number has one; not all do) and do the same thing as in step 1.
  4. Search Number Gossip for the whole number directly before and after the number you chose.  How are the search results different?  How are they similar?  Write a few sentences comparing and contrasting, as well as your thoughts as to why they compare the way they do.

 

 

 

The Math Of Roasted Potatoes

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Chefs in Great Britain have used “maths” to determine the best way to cook roasted potatoes.  See the magic formula here (courtesy of the Sun).

BADGING:

Watch the video and read the accompanying article.  Write a paragraph that explains the difference between the “traditional” way of making roasted potatoes and the “new, mathematical” way.  What changes were made and what is the advantage?  How did chefs determine that the new way was in fact “better”?

In a second paragraph, describe another food that you think might be improved using mathematics to aid in the preparation.  Describe a hypothesis you might have about how  that food might be improved using mathematics and why you think it would be an improvement.  Give a diagram if it helps your explanation.

She May Not Look Like Much, But She’s Got It Where It Counts, Kid.

 

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From Nerdist comes an infographic outlining the fastest space craft in the known universe — both real and fictional!  Check out the picture here! (Nerdist)

BADGING:

Look at the infographic linked above.

Pick out two of your favorite ships from the “relativistic” category.  Calculate their acceleration in m/s^2 (meters per second per second).  Compare those to a roller coaster in our very own Dollywood called the Tennessee Tornado, which reaches 3.7G.  How is it possible that a roller coaster achieves the same G-force as a space shuttle?

Then, pick out two of your favorite ships from the “faster than light” category (Note:  These are *all* fictional, as this type of travel is not yet possible).  Calculate their speed in miles per hour (You need to know that there are 1000 meters in a kilometer, and 1.62 km in a mile to do this calculation).  Please note that you may put your answers in scientific notation, and that you may not choose the Heart of Gold for this exercise.

*If you liked this badge, thank Eleanor for sharing the infographic with me!

 

 

What Are The Most Likeable Prime Numbers?

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So there’s this Twitter feed that tweets out the prime numbers…in order…on the hour…every hour.  Friend of Pre-Algebra.info David Butler analyzed the data from this Twitter feed to see which prime numbers were the most popular (via Likes and Re-Tweets).  See the results in his blog post.

BADGING:

`1.  Before you visit the blog post linked above, visit the Twitter feed that lists primes.  Out of the most recent 15 primes listed, which one is your “favorite”?  (You can decide how to interpret “favorite”.  Just decide which of the most recent 15 primes you like the best).

2.  Now visit Dr. Butler’s blog post where he analyzes the prime data.  Read the blog post and look at the data displays.  List at least four characteristics he noticed about the “most liked” primes, and give an example from the data to support each claim.

3.  Refer to your prime choice from question #1.  Does your choice fit any of the four categories from the second question?  Which ones?

4.  Hypothesize: Why do you think certain patterns or arrangements of primes are more “likeable” than others?  What might this have an impact on subjects like cryptography (internet passwords or even locker combinations)?

5.  Here’s a link to ten random ten-digit prime numbers?  Using what you’ve learned so far, which one do you think would be the “most liked”?  Explain your reasoning.

The Difference(s) Between Pro and Amateur Soccer Players

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With GPS data being used in a variety of sports, we have more data than ever to compare professional athletes.  More than just stats like goals and assists, we can actually measure distance traveled and top speeds.   The folks over at SportTechie have analyzed the data comparing amateur and professional soccer players.  Check out the differences between the players here.

BADGING:

Read the article linked above and answer the following questions (You may use a calculator, but show your work!).

  1.  The English Premier League leader in distance covered in the 2016/17 season was Tottenham’s midfielder Christian Eriksen, who covered an average of 11.92 kilometers per match.  If there are approximately 1.62 km. in a mile, how many miles did he average per match?  How does that compare to an average professional midfielder (data included in the article)?
  2. Let’s say an amateur attacker and a professional attacker are both racing towards the same spot on the soccer pitch.  Both players are 100 feet from the spot.  Calculate how far away would the amateur player be from that spot when the professional player arrived there, assuming both players ran at their top speed the entire time.  (There are 5280 feet in a mile, and use the speed data from the article linked above).
  3. An English Premier League season is 38 games long.  Over the course of a 38-game season, calculate how much farther a professional defender would run when compared to an amateur defender.

 

How Big is 52 Factorial?

52 factorial (or 52! in math notation) represents the number of ways a deck of cards can be shuffled.  It’s equal to 52 x 51 x 50 x 49 x 48 x … 3 x 2 x 1.  (Basically multiply all the whole numbers between 52 and 1 together.  The video above (from 14:20 to about 18:30) explains how large 52 factorial is.

BADGING:

BEFORE YOU WATCH THE VIDEO make a hypothesis about how long 52! is in seconds.  Is it an hour?  A day?  A week?  A year?  A lifetime?   A number of lifetimes?  Explain your reasoning in your hypothesis.  Make this a paragraph.

Then watch the video above (from 14:20 to 18:30 at least — watch the whole thing if you want to see some cool card tricks) and write a second paragraph that compares the actual length of 52! seconds to your hypothesis.

 

Major in Math If You Want A Career in Medicine

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A new article in the BBC suggests that mathematicians may be leading the fight when it comes to cancer research and discovering new things in the field of health and medicine.

Check out the article here:  http://www.bbc.com/news/science-environment-37630414

BADGING:

Read the article linked above.  Explain in a paragraph why mathematicians are so important in modern medical research.  In a second paragraph, explain the term “Datageddon” and why mathematicians need to be careful of it when conducting research.