Happy E Day!

In the United States, today is 2/7/18, matching the first four significant digits of e.

The next time that this date can be celebrated is July 2, 2018 (using the day/month/year format of abbreviated dates.) After that, we’ll have to wait until 27/1/82, or January 27, 2082. (Sadly, I knew about the number e back in 1982 but was then unaware of the day/month/year method of abbreviating dates, and so this day went unrecognized by me on January 27, 1982.)

Pythagoras vs. Einstein

Source: https://www.facebook.com/1698785500391556/photos/a.1698786577058115.1073741828.1698785500391556/1909519182651519/?type=3&theater

Digital Distraction

From the Chronicle of Higher Education: An Instructor Saw Digital Distraction in Class. So She Showed Students What She’d Seen on Their Screens.

Students get distracted in class, and all the shiny baubles that grab their attention are well chronicled. But what happens when students are presented with the greatest hits from their browsing history for an entire semester?

A graduate-student instructor at the University of Michigan at Ann Arbor, Meg Veitch, did just that. In an effort to keep students focused, she tracked all the times she had spotted them digitally wandering in class. She didn’t have access to their complete browsing history; rather, she used the low-tech method of writing down what she had spotted on students’ screens…

Ms. Veitch, who studies paleontology, presented her findings this week in a PowerPoint show for the class of roughly 160, which gave at least one student a chance to snap and share Ms. Veitch’s observations on Twitter:

My Favorite One-Liners: Part 78

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

One of my pet peeves is students who don’t give me a reasonable amount of time to grade their final exams and compute their grade for the semester. Usually, “reasonable” means “tomorrow morning.” (However, sometimes I have to give multiple final exams on the same day, and there just isn’t enough time in the day to compute grades for all of my classes in that amount of time.)

So, to head this off, I’ll announce to my students when they can expect me to finish grading the finals so that it’s safe to ask for their grade for the semester; usually the answer is “9:00 tomorrow morning.” And, to make sure that no one bugs me before then, I’ll give the following playful admonition:

Anyone who asks me for their grade before 9:00 tomorrow morning gets an automatic F in the course.

Was There a Pi Day on 3/14/1592?

In honor of Pi Day, here’s a bonus edition of Mean Green Math from two years ago.

Mean Green Math

March 14, 2015 has been labeled the Pi Day of the Century because of the way this day is abbreviated, at least in America: 3/14/15.

I was recently asked an interesting question: did any of our ancestors observe Pi Day about 400 years ago on 3/14/1592? The answer is, I highly doubt it.

My first thought was that $latex pi$ may not have been known to that many decimal places in 1592. However, a quick check on Wikipedia (see also here), as well as the book “$latex pi$ Unleashed,” verifies that my initial thought was wrong. In China, 7 places of accuracy were obtained by the 5th century. By the 14th century, $latex pi$ was known to 13 decimal places in India. In the 15th century, $latex pi$ was calculated to 16 decimal places in Persia.

It’s highly doubtful that the mathematicians in these ancient cultures actually talked to…

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The Perfect Geometrical Christmas Present

Behold, a three-dimensional calendar on the faces of a rhombic dodecahedron:

Source: https://www.wired.com/2016/10/gorgeous-12-sided-wooden-calendar-geometry-nerd-life/

Faculty Office Hours

Kudos to Arizona State University for making this public service announcement.

Pizza Hut Pi Day Challenge (Part 6)

On March 14, 2016, Pizza Hut held a online math competition in honor of Pi Day, offering three questions posed by Princeton mathematician John H. Conway. As luck would have it, years ago, I had actually heard of the first question before from a colleague who had heard it from Conway himself:

I’m thinking of a ten-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?

I really like this problem because it’s looks really tough but only requires knowledge of elementary-school arithmetic. So far in this series, I described why the solution must have one of the following four forms:

O , 4 O 2 , 5 8 O , 6 O 0,

O , 6 O 2 , 5 8 O , 4 O 0,

O , 2 O 6 , 5 4 O , 8 O 0.

O , 8 O 6 , 5 4 O , 2 O 0.

where O is one of 1, 3, 7, and 9. (The last digit is 0 and not an odd number.) There are 96 possible answers left.

Step 8. The number formed by the first eight digits must be a multiple of 8. By the divisibility rules, this means that the number formed by the sixth, seventh, and eighth digits must be a multiple of 8.

For the first form, that means that 8O6 must be a multiple of 8. We can directly test this:

816/8 = 102: a multiple of 8.

836/8 = 104.5: not a multiple of 8.

876/8 = 109.5: not a multiple of 8.

896/8 = 112: a multiple of 8.

For the second form, that means that 8O4 must be a multiple of 8. This is impossible. Let O = 2n+1. Then

8O4 = 800 + 10 \times O + 4

= 800 + 10(2n+1) + 4

= 800 + 20n + 14

= 2(400 + 10n + 7).

We see that 7 is odd, and therefore 400 + 10n + 7 is not a multiple of 2. Therefore, 2(400 + 10n + 7) is not a multiple of 4 (let alone 8).

For the third form, that means that 4O8 must be a multiple of 8. This is also impossible. Let O = 2n+1. Then

4O8 = 400 + 10 \times O + 8

= 400 + 10(2n+1) + 8

= 400 + 20n + 18

= 2(200 + 10n + 9).

We see that 9 is odd, and therefore 200 + 10n + 9 is not a multiple of 2. Therefore, 2(200 + 10n + 9) is not a multiple of 4 (let alone 8).

For the fourth form, that means that 4O2 must be a multiple of 8. We can directly test this:

412/8 = 51.5: not a multiple of 8.

432/8 = 54: a multiple of 8.

472/8 = 59: a multiple of 8.

492/8 = 61.5: not a multiple of 8.

In other words, we’re down to

O , 4 O 2 , 5 8 1 , 6 O 0,

O , 4 O 2 , 5 8 9 , 6 O 0,

O , 8 O 6 , 5 4 3 , 2 O 0,

O , 8 O 6 , 5 4 7 , 2 O 0.

For each of these, there are 3! = 6 ways of choosing the remaining odd digits. Since there are four forms, there are 4 x 6= 24 possible answers left.

In tomorrow’s post, I’ll cut this number down to 10.

A request to the athletic department

Even though I’ve had nothing but good professional relationships with the athletic department at my own university, I still think this is really funny.

Source: http://imgur.com/vJWkqhe

Shuffling cards

I really enjoyed this article concerning the mathematics of shuffling a deck of playing cards, justifying the claim that no two properly shuffled decks of cards have ever been the same: http://www.matthewweathers.com/year2006/shuffling_cards.htm