Computer Cracks 200 Terabyte Math Proof

Here’s a cute problem, called the Boolean Pythagorean Theorem problem. Here are the first few Pythagorean triples:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

9^2 + 12^2 = 15^2

8^2 + 15^2 = 17^2

12^2 + 16^2 = 20^2

7^2 + 24^2 = 25^2

15^2 + 20^2 = 25^2

10^2 + 24^2 = 26^2

20^2 + 21^2 = 29^2

18^2 + 24^2 = 30^2

16^2 + 30^2 = 34^2

21^2 + 28^2 = 35^2

12^2 + 35^2 = 37^2

15^2 + 36^2 = 39^2

27^2 + 36^2 = 45^2

9^2 + 40^2 = 41^2

27^2 + 36^2 = 45^2

OK, let’s have some fun with this. Let’s write every multiple of 5 (5, 10, 15, 20, 25, 30, 35, 40, 45) in boldface:

3^2 + 4^2 = {\bf 5}^2

6^2 + 8^2 = {\bf 10}^2

{\bf 5}^2 + 12^2 = 13^2

9^2 + 12^2 = {\bf 15}^2

8^2 + {\bf 15}^2 = 17^2

12^2 + 16^2 = {\bf 20}^2

7^2 + 24^2 = {\bf 25}^2

{\bf 15}^2 + {\bf 20}^2 = {\bf 25}^2

{\bf 10}^2 + 24^2 = 26^2

{\bf 20}^2 + 21^2 = 29^2

18^2 + 24^2 = {\bf 30}^2

16^2 + {\bf 30}^2 = 34^2

21^2 + 28^2 = {\bf 35}^2

12^2 + {\bf 35}^2 = 37^2

{\bf 15}^2 + 36^2 = 39^2

27^2 + 36^2 = {\bf 45}^2

9^2 + {\bf 40}^2 = 41^2

27^2 + 36^2 = {\bf 45}^2

For nearly all of these equations, there is one number that’s in boldface and one that’s not. However, there’s one that is all in one typeface: {\bf 15}^2 + {\bf 20}^2 = {\bf 25}^2.

So here’s a question: is it possible to divide the integers so that every Pythagorean triple (not just the small ones listed above) has at least one number in boldface and another that’s not?

This May, it was proved that it’s impossible. The proof is very brute-force (from https://cosmosmagazine.com/mathematics/computer-cracks-200-terabyte-maths-proof):

The team found all triples could be multi-coloured in integers up to 7,824. As soon as they hit 7,825, it became impossible.

But to prove a solution doesn’t exist, you need to try all possibilities. There are more than 10^{2300} ways to colour all those integers, so the scientists used a few mathematical tricks to reduce the number of combinations to trial to just under one trillion.

Two days later, with 800 processors at the University of Texas Stampede supercomputer crunching all possibilities in parallel, the team had their answer – no.

There is no way to colour the integers 1 to 7,825 in a way that leaves all Pythagorean triples multi-coloured, the team reported in arXiv.

I had to read this news article a couple of times to appreciate this: a supercomputer ran for two days on a supercomputer (without parallelization, computation time was 51,000 hours), producing an output file of 200 terabytes, comparable “to the size of the entire digitized text held by the US Library of Congress.” Wow.

Sphere Packing Solved in Higher Dimensions

I enjoyed reading this bit of mathematical news: https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/

The opening paragraphs:

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

Mathematicians have been studying sphere packings since at least 1611, when Johannes Kepler conjectured that the densest way to pack together equal-sized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores. Despite the problem’s seeming simplicity, it was not settled until 1998, when Thomas Hales, now of the University of Pittsburgh, finally proved Kepler’s conjecture in 250 pages of mathematical arguments combined with mammoth computer calculations.

Do’s, Don’ts for Parents to Help Teens Build Math Interest and Success

I really enjoyed reading this article: http://www.usnews.com/education/blogs/high-school-notes/2016/04/11/dos-donts-for-parents-to-help-teens-build-math-interest-and-success

A summary:

  • Don’t project negative feeling toward math onto teens
  • Do talk to teens and teachers about what’s being taught in math class
  • Don’t be too quick to hire a tutor for struggling students
  • Do support students with the right tools

I recommend the whole article and the references therein.

Statistics and percussion

I recently had a flash of insight when teaching statistics. I have completed my lectures of finding confidence intervals and conducting hypothesis testing for one-sample problems (both for averages and for proportions), and I was about to start my lectures on two-sample problems (liek the difference of two means or the difference of two proportions).

On the one hand, this section of the course is considerably more complicated because the formulas are considerably longer and hence harder to remember (and more conducive to careless mistakes when using a calculator). The formula for the standard error is longer, and (in the case of small samples) the Welch-Satterthwaite formula is especially cumbersome to use.

On the other hand, students who have mastered statistical techniques for one sample can easily extend this knowledge to the two-sample case. The test statistic (either z or t) can be found by using the formula (Observed – Expected)/(Standard Error), where the standard error formula has changed, and the critical values of the normal or t distribution is used as before.

I hadn’t prepared this ahead of time, but while I was lecturing to my students I remembered a story that I heard a music professor say about students learning how to play percussion instruments. As opposed to other musicians, the budding percussionist only has a few basic techniques to learn and master. The trick for the percussionist is not memorizing hundreds of different techniques but correctly applying a few techniques to dozens of different kinds of instruments (drums, xylophones, bells, cymbals, etc.)

It hit me that this was an apt analogy for the student of statistics. Once the techniques of the one-sample case are learned, these same techniques are applied, with slight modifications, to the two-sample case.

I’ve been using this analogy ever since, and it seems to resonate (pun intended) with my students as they learn and practice the avalanche of formulas for two-sample statistics problems.

Engaging students: Radius, Diameter, and Circumference of a Circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Zacquiri Rutledge. His topic, from Geometry: radius, diameter, and circumference of a circle.

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There are many ideas about how to introduce students and have them study the relationships between the radius, diameter and circumference of a circle. However, one of my favorites has always been the month long project assigned to students at the beginning of class. On the very first day of class, the teacher is to assign the students their project. The instructions of this project are for each of the students to find and measure ten different round or circular objects around their home. The students will need to measure the length around the object (the circumference) using a piece of string and a ruler (the teacher might explain to the students or give an example so they know how to do this), the length from one side of the object to the other side passing through the middle (diameter), and the length from the center of the object to the outside (radius). If the students already know what these terms are called that is okay. However, the teacher should avoid explaining these terms until later.

Then a month later, the students are to bring their findings to class. At this point during the class the teacher will have begun her segment of lessons about circles and the various properties of circles. By now the students should have a good idea what the terms radius, diameter, and circumference mean. So the day the students bring in their work, they will be given the following chart, originally designed by the University of Illinois. From here students will slowly begin to fill in their charts with the information they gathered. Once completed students will then begin finding the ratios between diameter-radius and circumference-diameter and recording them. Finally at the bottom, students will find the average of their ratios from the last two columns. Once all of this data is completed, the students should have found that the diameter and radius share a ratio of 2-1 since the diameter is twice the radius. The last column should have produced something close to an average of 3.14159265359 or better known as pi (). Not only will this help students understand that pi is not just a number, but it will also help them to know where it comes from and its importance. From here the teacher would be able to lead into a lesson about some of the other uses of pi and how they all relate back to the relationships between radius, diameter and circumference.

pichart

 

 

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Radius, diameter, and circumference are very important in many topics beyond their definitions. For instance, later on in the geometry course students will talk about the area of circles. Even though the students might have learned how to find the area of simple polygons such as triangles and quadrilaterals, finding the area of a circle is different because of the use of pi. To find the area of a circle, students have to recall the relationships between the radius, diameter, and circumference in order to understand how the area of a circle uses those relationships. Another example of how they are used is in pre-calculus. In pre-calculus students will talk about the unit circle, a circle with a fixed radius of 1 unit. Using the fixed radius of 1, students will discover that the length around (circumference) the unit circle is 2π. This 2π is important because it can be broken into pieces, called radians, and used to help measure Sine, Cosine, and Tangent at certain radians around the circle. Learning about sine, cosine, and tangent opens up even more things for the students, such as trigonometry and calculus. However, no matter how advanced the mathematics become, they always relate back to the simple concepts of the radius, diameter and circumference of a circle and their relationships.

 

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Radius, diameter, and circumference is a topic that has been talked about and used dating back to 2000 B.C. But, what has it actually been used for all this time? How about architecture? Think about massive constructs such as the Theatre of Ephesus in Rome, Italy. Even though the theatre is not a full circle, look at how each of the seats are evenly placed from the stage. This is because when it was designed, the architect likely used the radius and circumference to accurately plot how far each seat needed to be placed in order to be the same exact distance from the stage as everyone else in their row. Even though only half a circle was used for this theatre, the circumference and radius would have been used to find the ratio pi in order to get the area of how much space was allowed for seating.

Another great example of circumference being used is in the invention of the clock. The clock originated as a sun dial, which would use the sun to cast a shadow, which would tell the time of day. These sun dials date back as early as 3500 B.C. However, in 1583 Galileo found a way to use a pendulum to create a clock that always followed the same length of time (Clock). This is important because not long after the first clock was born, so was the circular face of a clock. The face of a clock has the numbers 1-12 on it, each one evenly spaced around the edge of the clock. By using the circumference of any size of circle, the person building the clock would know just how far to space out each of the numbers, giving each hour the same amount of time between them. If even one of the numbers were off on the clock, the time would be off. Also, it can be seen that on modern clocks, the minute hand always stretches the radius of the clock. By stretching out the minute hand on the clock, the designer of the clock can create evenly spaced notches on the face using the circumference, in order to have the minute hand indicate the minute of the hour.

One final example is the use of radius in war, or more specifically the invention of the radar. Radar was originally being experimented with by German physicist Heinrich Hertz in 1887. He had discovered that certain materials allowed radio waves to pass through them, while others reflected them. In 1890, Nikola Tesla realized that large objects could reflect large enough radio waves to be detected. By harnessing this idea, pulse radar would come to be introduced into United States in 1925, and later used in the British Air Force to defend against German air raids during WWII (Science). The reason radar works, however, is because the system has a set radius in which it can detect radio waves. Once the radar system sends out a radio wave, if it does not reflect back within the radius of the detection system, then the radar will not pick up on anything. The system measures the distance by measuring how long it takes for the radio wave to return to the system after it is sent out and comparing that time to radius of detection. This allowed not only military to defend against air attacks, but it was commonly used during naval combat to defend against submarines as Germany used their U-Boats to attack several American and British naval ships during WWII, as well as WWI before the invention of radar.

 

References:

“Circumference and Pi.” Circumference and Pi. N.p., n.d. Web. 08 Oct. 2015.

“Clock a History – Timekeepers.” Clock a History – Timekeepers. N.p., n.d. Web. 08 Oct. 2015.

“Science Explorations: Journey Into Space: Radar and Sonar | Scholastic.com.” Science Explorations: Journey Into Space: Radar and Sonar | Scholastic.com. N.p., n.d. Web. 08 Oct. 2015.

 

 

 

 

Engaging students: Defining the terms acute triangle, right triangle, and obtuse triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Taylor Vaughn. Her topic, from Geometry: defining the terms acute triangle, right triangle, and obtuse triangle.

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How can this topic be used in your students’ future courses in mathematics or science?

As soon as you think triangles are gone, they are not. In pre-calculus you will address these triangles again, but in a different outlook. In pre-calculus you will notice patterns associated with sin, cos, tan and the different triangles, acute, obtuse, and right. Also there is a cool theorem called Pythagorean Theorem, a2 + b2 =c2, where a and b are the legs and c is the hypotenuse. This theorem you will forever use, no matter how up in math you get. In calculus right triangles are used for trig substitutions.  Trig substitution is instead of using the number, you use sin, cos, tan, sec, to solving different equations. So triangles you want to always remember because in math everything is linked together amd almost everything is a pattern.

 

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This semester I have the pleasure of working at the Rec. Being a supervisor for intramurals causes me to a lot of the behind the scenes work that I didn’t know happened. One is turning a patch of grass into a football field. I know you probably thinking what does this have to do with anything, but I actually used 3-4-5 triangles, right triangles, to draw the field.  So when laying down the basics of the field we had to mark of 15 yards from a fence so that participants would hurt themselves. Then I placed the stake at that spot. Then we tied twine around the stake and walked down 100 yards and placed a stake. Then wrapped a new piece of twine to the new stake and measured of 40 yards for the width (measurements comes from NIRSA handbook, which are the rules we go by for flag football). Then did the same for the other side to get a rectangle of a length of 100 yards and with of 40 yards. When I saw this paint can, it then hit me that we had to actually paint this. SO my question was “How am I supposed to get straight line?” Well to my shock, my boss pulls up the measuring tape and said “a 3-4-5 triangle!” Who knew! So for the first corner we measured down the twine 3 yards and then 4 yards going into the field and placed a stake. Then we had to twine the two together measuring to see if it was 5 yards. If it wasn’t we had to keep moving the stakes till they were. Once it was it was for sure that the twine was straight and you could use the paint machine and just push along the line. You do this process and until all the lines are done, even for the yard marking lines , like the 20 yard line, and 40 yard line, that you see on the field. Just as shocked as I was, I bet students will be too. Here is a video to show what I am saying so if it is a little confusing the students will have a visual. Or definitely and visual you could do to show this.

 

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

One cool activity I found was an online game called Triangle Shoot, where you had to classify the triangles. The game has a lot of floating triangles and on the bottom of the cursor it says what triangle you need to click. Before you start the game, it gives definitions and pictures of the triangles before starting. I played it myself and actually found it fun. For me, the timed mode was more fun due to the fact as time got closer to 0 the more pressure I felt trying to beat my previous score. And since the shapes are floating you try to click them before they float away. I also liked that the shapes are not always facing the same way, some are rotated on its side or flipped, which made it a little more difficult. It also calculates a percentage and tells you how many you got wrong and right. The only thing I wish it did was break down the hits and miss according to the triangle that way students know what triangle that understand ad don’t. I really thought this was a fun activity after introducing the vocabulary. The website is actually a good tool for students to practice what each triangle is and how they differ. Even if a school doesn’t have computers that students could actually try this in class, it is something that students could use as a practice. Also the game has a mode where you can do equilateral, isosceles, and scalene triangles. http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/triangles_shoot.htm

 

References

Ricalde, Paul. “3-4-5 Method, How to Get a Perfect Right Angle When Building Structures.” YouTube. N.p., 28 Mar. 2013. Web. 7 Oct. 2015.

“Triangle Shoot.” Sheppard Software. N.p., n.d. Web. 7 Oct. 2015.