Problem: "Puzzle math"


I've added the following puzzle to the Problems page: For the 8 squares below (corners aren't included) squares are adjacent up/down/left/right/diagonally. Fill in each square with a number from 1 through 8 (one time each) so that adjacent squares don't contain consecutive integers. I found the problem here. You can reason it out logically but I ended up using graph theory to get the answer.

Lots of stories this week:

  • A MUST READ article by John Bohannon: "I Fooled Millions Into Thinking Chocolate Helps Weight Loss. Here's How". From the article, "It was, in fact, a fairly typical study for the field of diet research. Which is to say: It was terrible science. The results are meaningless, and the health claims that the media blasted out to millions of people around the world are utterly unfounded." and the key to generate bad conclusions from good data is "...If you measure a large number of things about a small number of people, you are almost guaranteed to get a “statistically significant” result. Our study included 18 different measurements—weight, cholesterol, sodium, blood protein levels, sleep quality, well-being, etc.—from 15 people. (One subject was dropped.) That study design is a recipe for false positives.". And the p-value is instrumental in the deception, "It’s called p-hacking—fiddling with your experimental design and data to push p under 0.05—and it’s a big problem. Most scientists are honest and do it unconsciously. They get negative results, convince themselves they goofed, and repeat the experiment until it “works”. Or they drop “outlier” data points.". This article is a great resource if you teach statistics.
  • Magnus Carlsen won a 3 board blindfold (with clock) exhibition. The video is posted on Chessbase.
  • Caruana and Nakamura earned their place in the upcoming Candidates tournament to determine the next challenger for the World Chess Championship by taking the top two places at Khanty-Mansiysk 2015.
  • The editor in chief of the Lancet, one of the world's most prestigious medical journals has stirred up some controversy. "Dr. Horton recently published a statement declaring that a lot of published research is in fact unreliable at best, if not completely false. “The case against science is straightforward: much of the scientific literature, perhaps half, may simply be untrue. Afflicted by studies with small sample sizes, tiny effects, invalid exploratory analyses, and flagrant conflicts of interest, together with an obsession for pursuing fashionable trends of dubious importance, science has taken a turn towards darkness.. Yes, statistics isn't really math.
  • Forbes has one of those REALLY annoying posts by someone who really lacks basic knowledge about what they're writing about--something all too common in mainstream media where even robots now generate worthless content. Although posed as a question "Should We Stop Teaching Calculus in High School?" the author is clearly saying "yes". "The list of high school math courses in the U.S. hasn’t changed for decades. My daughters are taking the same courses I took long ago: algebra, geometry, trigonometry, and calculus. These are all fine subjects, but they don’t serve the needs of the 21st century.". But the author goes on to say, "...the vast majority will never use calculus again. And those who do need it – future engineers, physicists, and the like – can take it in college.". So the courses do serve the needs of the 21st century. The author makes the point that we are awash in data today so the author asks, "What math courses do young people really need? Two subjects are head-smackingly obvious: computer science and statistics.". Huh? Who believes computer science is math? And anyone reading this blog knows (e.g. see here and here) that statistics isn't math either. Yes, theoretical stats is basically analysis but the statistics he's talking about (confidence intervals, p-values, etc) isn't. Whether (see the links) you want to look at schools having a "department of math and statistics", or that bigger schools have a separate statistics department, or that AMSTAT news says statistics isn't a subfield of math, or as is mentioned in the second link that statistics books and teachers have been presenting p-values incorrectly. If you believe stats is math then please explain what other branch of math teaches you the wrong way to do something as has been done with p-values? There is a different reasoning process for stats. But back to the article. With respect to statistics, "Most high schools don’t offer either one. In the few schools that do, they are usually electives that only a few students take.". Let's mention that statistics is covered in Common Core. The Common Core Standards are posted here and this government site notes, "The recently adopted Common Core State Standards for Mathematics (CCSSM) contain a large amount of statistics in the middle and high school grades and some at the elementary school level.". Not being a math teacher, this author is unaware how much things have changed: more statistics, mathematical proofs in geometry have been largely removed (I even had to teach probability(!) in my geometry classes), and 4 ways to subtract (which has confounded parents) are some noticeable changes. Assuming that because his daughters are taking the same courses he took decades ago means there hasn't been any change in content is, at best, sloppy journalism. To be clear, let me agree that computer courses in high school would be great. Python is such a natural choice that could be useful--but you don't sacrifice core math classes for that; you eliminate or consolidate less relevant courses to make room for it. The removal of most proof content from geometry is tragic because proofs are the essence of mathematics. As Alfred Renyi said, "A mathematician is a device for turning coffee into theorems". The author opting for computer science and stats as math classes while neglecting discrete math is, I suspect, based in the ignorance of not knowing discrete math is the math of computer science. The author continues, "Convincing schools to give up calculus won’t be easy. I imagine that most math educators will scream in protest at the mere suggestion, in fact. In their never-ending competition to look good on a blizzard of standardized tests, schools push students to accelerate in math starting in elementary school, and they offer calculus as early as the tenth grade. This doesn’t serve students well: the vast majority will never use calculus again.". There's some truth in here but since he hasn't taught high school he can't properly interpret what's happening. High schools get awarded numerical scores according to a formula for "performance" (which often get translated into star ratings for the school). Admin look at how they can increase scores (to make their performance look better). More students taking AP exams means a higher score is given, regardless of how poorly the students do. That's why school admins get teachers to encourage students to sign up for AP classes and that's why schools often pay for the student to take the AP exam--it's an easy way to raise the school's score. The consequence is that it's commonplace for students who struggle with fractions to be taking AP Calculus. But no matter, just require a graphing calculator to give students a chance. The child feels smart, the parents feel proud, admin performance improves, and some business makes a lot of money selling expensive calculators when you can buy a laptop computer for $200. The only problem is the child still has poor math skills that they'd be put to shame by a typical student from another country at a lower grade level that has a fraction of the resources but has parents making sure kids learn multiplication tables and basics and not giving them a calculator to use as a crutch at lower levels. But the issue isn't about producing quality in the US, so it's no surprise we never get it. They're looking to maximize performance under the rules they've been given so wasting taxpayer money on improving the school's score gets the admin credit for improving school quality even though no real quality has taken place. Same thing with attendance. Some schools have poor scores due in part to poor attendance. Since that poor attendance is, in cases, predictable (before Christmas break) schools often have incentives (such as a drawing for a free computer) for students who attend. This expenditure of taxpayer money has nothing to do with quality. There's so much to criticize in this article but you get the idea. His argument that "here’s a simple fix: get rid of high school calculus to make way for computer programming and statistics" is nonsense.
  • The controversy of why women don't perform as well as men in chess, which I first raised here, continues with a new study (authored in part by women). The data itself is interesting: "“Hard” sciences such as physics and statistics on average have a larger gender gap than social sciences and humanities – no surprise there. However, this is only part of the story. According to 2013 data from the National Science Foundation in the United States, there is large variation within each category: whereas women earned only 19% of PhDs in physics and 18% in computer science, they earned no less than 54% of PhDs in molecular biology – amounting to gender parity. Within the humanities and social sciences, those numbers ranged from 78% in art history and 72% in psychology to a dismal 27% in philosophy and 28% in music theory and composition.". I think the conclusions drawn off the data are suspect, to put it lightly. "The key claims of Leslie and Cimpian’s paper are: 1. that fields vary in the degree to which its practitioners believe that innate ability (“genius” or “brilliance”) is required for success; and 2. that society often promotes the notion that men have greater innate abilities than women.". If women were doing well in those fields then I don't see how they would be "bluffed" out of continuing. It makes more sense to me that they weren't doing well, didn't like it, found their "passion" somewhere else (as jobs opened up in other more desirable fields), etc.. The fact is that women used to hold a large percentage of computer science job. The claim that women would somehow give up pursuing a job in these lucrative white collar fields  because  society has told them men have greater innate abilities seems demeaning to women and contrary to the general rise of women in the workforce over the past 40 years. Women have faced harassment in many areas and haven't quit. More believable to me is that the radical change in the field took computer science out of their comfort zone: computer science today is so much different than back in the 80's. Note also that the study separates Statistics from math (as it should) and the resulting percentages for the two are quite different.
  • The chronotope blog comments on the (included) recent John Oliver video on student test in high schools. Make sure to check out the video!
  • Ravens guard John Urschel analyzes the extra point rule change in football.
  • The Hindu notes on the passing of chess legend Anand's mother. Grandmaster R.B. Ramesh has a fitting quote, "Without Anand it’s tough to imagine Indian chess. Without his mother, it’s tough to imagine Anand”.
  • CinemaBlend posts on a Tobey Maguire starring as former World Chess Champion Bobby Fischer in "Pawn Sacrific". Check out the movie trailer.
  • Hundreds of SATs go missing. If they aren't found soon the teenagers will have to take them again. Let's hope nobody has their college plans derailed.
  • reports  "Wisconsin may be the first state in the country to certify teachers who don't have bachelor's degrees under a provision put in the state budget....Under the change, anyone with relevant experience could be licensed to teach non-core academic subjects in grades six through 12. They would not need a bachelor's degree and they could even be a high school dropout.". This doesn't sound like a good idea.


Sagetex: Indefinite Integrals 3/4


I've added two more indefinite integrals to the Sagetex: Integrals page.

Here are some issues that caught my eye over the last week:

  • The NY Times posts on the tragic death of Nobel prize winning mathematician John Nash and his wife on the NJ Turnpike.
  • The Khanty-Mansiysk 2015 chess tournament continues here with Caruana leading the pack with +1 and only two rounds to go.
  • reports on how a principal was "released" after making racially charged comments during graduation exercises. The posted video lets you judge for yourself.
  • Al-Jazeera reports on how Texas is punishing students for skipping school. From the article (video posted, too): "In Texas, truancy is a Class C misdemeanor that carries a possible $500 fine and the threat of jail time if fines go unpaid when a child turns 17. The law has captured the attention of advocacy group Texas Appleseed, whose research found disabled, poor, and black and Latino students are most often sent to truancy court.".  The poor mom is working 2 jobs and now has extra financial stress trying to pay for her truant son. After watching the video this story describes the pathetic state of many of today's kids and their parents who want to be their friends. Why does the child decide whether he goes to school?!?
  • Stitz Zeager Open Source Mathematics has a variety of free, good quality math ebooks as well as the LaTeX source files.

LaTeX: PDF and tex file output


In an earlier post I mentioned how the output from a Sage session could be saved to a file--you just needed to know the path. As I alluded to, it's possible to do the same through Sagemath Cloud using sagetex, too. That means you can create a pdf and tex file at the same time. But why?

If "because it's fun" doesn't make your list, I've come up with a more plausible scenario. The original Sage Test Template provided a starting point for designing a randomized test. The result was a test along with the answer key in a 3 page pdf. But suppose you don't want the PDF to contain the answer key? It's possible to create a PDF for the test while at the same time producing the tex file for the answer key. Although cumbersome, it can be done. You'll need to create a file

f = open(r"SMCtestAnswers.tex",'w')

As with the earlier post, the file is created if it doesn't exist already. Unlike the that example there is no path given. Once the file is created it's just a matter of filling it with the preamble in order to run and then the answers using the choice of random variables from compilation. So the setup and first solution looks like this:

f.write(r"\usepackage{amsmath, amsfonts, amssymb}")
f.write(r"{\large Answers}")
f.write(r"$x=%s$, and $x=%s$"%(a,c))

And you can see why cumbersome is a kind description. Each line followed by a newline character (it didn't work as one line) and when it's time for the solution we insert the appropriate variables. After compiling you get a pdf for the test and you'll have to go check your files to find the newly created SMCtestAnswers.tex. Compile that file to create a pdf just for the answers.


Mission accomplished. I've posted the "proof of concept" file on the Handouts page.

Here are some issues that caught my eye this week.

  • The Khanty-Mansiysk 2015 chess tournament will seed the top two finishers to be in a tournament which will select the next challenger for World Champion Magnus Carlsen. As such you can find many top players competing. Caruana leads after 4 rounds. Tomorrow is a day off. The tournament can be followed at the site here or through Chessbomb (sidebar).
  • Tragedy in Jacksonville where according to this article in the St. Augustine Record, "A girl was left with a gunshot wound through both cheeks. Another girl was struck in the head by a shot. Both survived, but for a moment Thursday, a routine school bus ride home turned into a ride of terror on Jacksonville’s Westside.".
  • Students behaving savagely: ZeroHedge has the story of 50 teenagers beating an adult into the ground: "Richard Fletcher was beaten by a gang after asking two girls to stop fighting on his truck in Dundalk, Maryland. Man, 61, left with horrific injuries and facing $400,000 medical bills after near fatal attack by pack of FIFTY teens, including girls, when he tried to break up a fight. A 17-year-old boy has been charged as an adult for his role in beating a 61-year-old alongside a group of approximately 50 other teens in Baltimore, Maryland, on April 22.". Five students arrested at the time of the story.
  • In a post on Thomas Jefferson high school we saw people suing the school because "...the numbers [of black students] could surely be higher than they are now.". Now from the Washington Post we have, "More than 60 Asian American organizations filed a complaint (see below) with the federal government on Friday alleging that Harvard University discriminates against Asian Americans in the admissions process and calling for an investigation.". The complaint is posted at the link above. Sputnik news has these statistics: "A coalition of 64 organizations is citing research indicating that Asians need to on average score 140 points higher than white students on the SAT, 270 points higher than Hispanic students and 450 points higher than African-American students to equal their chances of gaining admission to Harvard. The exam is scored on a 2400-point scale.". It seems like the reverse discrimination laws to help "level" the field for Asian-Americans is now legal policy to discriminate against them for performing better than other racial groups.

Understanding "The Test"


The post Understanding the Prediction explained the mathematics behind Richard Wiseman's brilliant magic trick that's posted on his Quirkology site. It's a great way to introduce graph theory to your class.

Richard Wiseman's video "The Test" is a similar type of magic trick that can be explained using digraphs and, once again, is a great way to introduce a class to digraphs. I've added an explanation of "The Test" to the Other page.

Here are some thngs that caught my eye last week:

  • Discover blog has an article, "The Purpose of Harvard is not to Educate People". From the article, "Don’t believe me? Here is the test: when was the last time Harvard made a senior tenure offer to someone because they were a world-class educator, rather than a world-class researcher? Not only is the answer “never,” the question itself is somewhat laughable.".
  • reports on a "...former interim director of special services for the Brick Township School District, failed to reveal a 1990 conviction on heroin and cocaine charges on his job application with the district, the Ocean County Prosecutor’s Office has confirmed. Morgan, 65, was charged Thursday along with Brick Schools Superintendent Walter Uszenski and Uszenski’s daughter, Jacqueline Halsey, in a scheme that supplied Halsey with full-time day care for her preschool child paid for by the Brick Township schools, with official misconduct and theft by deception, said Al Della Fave, spokesman for the Ocean County Prosecutor’s Office.....According to a 1989 report in the New York Times, Morgan was arrested and charged with selling cocaine on five occasions, in amounts ranging from a half-ounce to more than 3 ounces, to undercover detectives assigned to the Brooklyn District Attorney’s investigation unit.Morgan -- who taught English to 9th and 10th graders in a special education program at Canarsie High School -- also was charged with possession of cocaine, as well as with conspiracy to sell heroin, in which he is said to have agreed to travel to Thailand to buy heroin for undercover agents posing as drug dealers. The heroin was supposed to be brought into this country concealed in disposable diapers, the authorities said.Morgan later was convicted of felony drug charges, though a follow-up article in the New York Times does not specify the exact counts. Morgan, who had worked for the school for 20 years, was fired but later won a civil case against the schools over the firing, despite his conviction.". So you've got a school official arrested and charged with selling drugs multiple instances (and the details are such public knowledge that it made the NY Times) and was teaching kids. So the question becomes "Was a background check ever done?" If it was, "How did it miss such well known information?". Also ask yourself what it says when Morgan was hired at the "request and recommendation" of someone still working in the educational system.
  • (Stephen) "Colbert Fune $800K in Grants for SC teachers".
  • Huffington Post has an interview with 2010 Fields Medal winner Cedric Villani.

Sage Essentials: Writing to a file


I've added some information on writing to a file while using Sage installed on your computer. It's on the Sage Essentials page. Essentially you start with a line like

f = open(r"/home/<rest of path>/MyFile.tex",'w')

while filling in the particulars of your path on your computer and after including some code like that shown above, Sage will create the file for you (regardless of whether it had existed before). If you use Sage, the chances are you can think of some times when you'd want to output data/results to a file. Whereas a text file (or .csv file) is probably what most people are used to, the TeX users amongst us will be more interested in .tex files to get the beautiful typesetting of LaTeX.

But remember, Sage can be run through LaTeX using the sagetex package, so this raises the possibility of running LaTeX and, in addition to creating the PDF file, creating multiple .tex file as well. Is there a use for that? From this TeX Stack Exchange post, the answer is yes. And since multiple PDFs are not an option unless the small changes between documents are "predictable" (e.g. change of names in a letter), .tex file output might prove useful as well. More on that later after I have some time to tinker with the idea. Here are some stories that caught my eye this week:

  • Remember the outrageous charges of sexism against Nigel Short? Peter Long, from the Malay Mail Online has a good opinion piece on this and other chess related issues here. He has a busted link  " a very well-written article by Tarjei J. Svensen entitled Susan Polgar and Nigel Short at War which shows that the underlying (real) issue written about by Short in the New in Chess column is certainly not sexism (let alone race, which was sadly thrown into the mix by some without any thought).". There's so much going on here! You can find that Svensen article posted on Chess24. Also, a reader has found a piece in the Guardian of the study that Short was, most likely, citing.
  • One News Now reports some disturbing news: "South Carolina Education Department Chief Operating Officer Elizabeth Carpentier issued a warning to parents that they could be locked up behind bars for 30 days if their children missed even one day of testing, according to South Carolina Parents Involved in Education member Tamra Hood as stated by Breitbart News.Carpentier didn’t stop there, reportedly threatening any groups or organizations involved in encouraging parents to object to Common Core testing with charges of aiding and abetting a crime.School officials made it loud and clear that students and parents have no say in the matter and must abide by the state’s dictates whether they like it or not."
  • The Washington Post has a piece in which a teacher gives a well articulated argument (backed up with specific examples) of the problems in common core exams: "I want to talk about the test itself. It is a fundamentally flawed tool that will only debase the good work we teachers do in the classroom, the work that districts do in designing and implementing quality curriculum, and the work our students do in learning to become enlightened critical thinkers....First things first, one of the most disturbing trends that I have found examining this year’s and last year’s (released) tests is a shift in thinking toward a kind of intellectual relativism. In other words, any claim that a student makes is correct if he or she substantiates it with some evidence. On the surface this doesn’t sound terribly problematic, but when you start to examine some of the anchor papers, the dilemma with this vein of thinking becomes shockingly apparent. The truth is, not all claims are correct and not all evidence is created equal. Making a feeble claim and using evidence out of context to support that claim is an all too common occurrence on these tests....Another disturbing pattern that emerges as one reads the anchor responses for the ELA is what I call “The Easter Egg Hunt.” When it comes to short answer questions in particular, the question that is actually being posed rarely matches the answer required. The wordier the written response, the more likely it is that the student will stumble upon the correct answer, find the decorative egg. (Strategy!) Time after time there is a clandestine condition that must be met in order for an answer to get full credit – “Magic Words.” As my scoring instructor illustrated, it’s kind of like tossing all of the words into a bucket and looking for certain key phrases or ideas to float up to the top."
  • ZeroHedge has the story of a Texas college teacher who snapped and failed the entire class via e-mail, "While we don’t know if any of the above factored into the following story, what we do know beyond a shadow of a doubt is that Texas A&M Professor Irwin Horwitz was sick and tired of the students in his Strategic Management course and so he did what many a college professor across the country has at one time or another dreamed of doing: he failed the entire class. Here’s an excerpt from the e-mail he sent to his students: "Since teaching this course, I have caught and seen cheating, been told to 'chill out,' 'get out of my space,' 'go back and teach,' [been] called a (DELETED: see the link for the proper description) to my face, [had] one student cheat by signing in for another, one student not showing up but claiming they did, listened to many hurtful and untrue rumors about myself and others, been caught between fights between students. None of you, in my opinion, given the behavior in this class, deserve to pass, or graduate to become an Aggie, as you do not in any way embody the honor that the university holds graduates should have within their personal character. It is thus for these reasons why I am officially walking away from this course. I am frankly and completely disgusted. You all lack the honor and maturity to live up to the standards that Texas A&M holds, and the competence and/or desire to do the quality work necessary to pass the course just on a grade level…. I will no longer be teaching the course, and all are being awarded a failing grade.""
  • MetroUK posts a picture of work submitted by a student trying to put one over their teacher, "The teacher clearly didn’t appreciate the cunning blend of the words ‘true’ and ‘false’.Genius as it is, trying it out on every answer was always destined to fail – although he might have been lucky if he’d only slipped it in just once or twice."