The Sum of our Knowledge

This problem, along with a couple of others, was inspired by the work of John H. Conway* and appeared in  the October 15, 2020 edition of Quanta Magazine . Let a, b, c, d, e, f, g, h, i, j be the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in some order. Each digit appears exactly once. (For those that want a more precise mathematical phrasing, there is a one-to-one correspondence between the set {a, b, c, d, e, f, g, h, i, j} and the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, or better yet, there is a bijection between the two sets.)  We form other integers by concatenating digits, so for example, 'ab' is a two-digit number where 'a' is the tens digit and 'b' is the ones digit. If abcdefghij is a 10-digit number with the following properties: a is divisible by 1 (OK, that much is obvious) ab is divisible by 2 (So, b must be even) abc is divisible by 3 (Do you know the test for divisibility by 3?) abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg ...

COME ON DOWWWWNNNN!!!!!!   You’ve been chosen to be a contestant on Let’s Make a Deal!   You’re now standing in front of three doors -- behind one of these doors is a car, and behind each of the other two is a goat. Monty, the host, says, “Pick a door! If it has the car behind it, you’ll drive it home today!”  You choose door #1, and feel really good about your choice.  The host then opens door #3,  which has a goat behind it, and asks “Do you want to switch doors?”  Question: What is the probability that you will win the car now that you know there was a goat behind door #3? Should you switch? Assumptions: Before the show, the car was equally likely to be placed behind any of the three doors. Once you make your selection of door #1, Monty will choose to reveal what is behind one of the other two doors, but will definitely not show you where the car is. Update: 11/19/20: read below for the solution.

I was honored earlier this month to be recognized by Stephen Wolfram as one of the recipients of the Wolfram Innovator Award at the annual Wolfram Technology Conference.   (Here is the official announcement of the Award Winners . ) This award is given annually to educational and scientific pioneers that use Wolfram’s Mathematica ™ software to bring the future to today.   The awarded project was the result of a large number of people at the University of Minnesota working to lower the cost of college attendance by replacing expensive textbooks with technology better suited to the needs of 21 st century students. In addition, this technological innovation has improved the way we deliver instruction at MathCEP . The project, now known as the Minnesota Online Learning System (MOLS), delivers assessments to students using the Mathematica platform. It replaced a publisher’s automated homework system initially, and then expanded to house the University of Minnesota’s math placeme...

Here's a puzzle that was first passed on to me by Al Lippert, who spent years providing the "Problem of the Day" at MathPath .  What goes after the last equals sign? Hints: The answer we're looking for isn't 80.  And the equalities are all true! Update: 11/6/20: read below for the solution.

  As COVID ramped up last spring, in-person classes came to an abrupt halt.  Many of us had just a few days to transition into online teaching, and very little ( if any ) formal training.  In fact, most of us had never even taken an online course ourselves. We lacked any model for constructing our courses in the online environment, and we did the best that we could to just teach our students something… anything! We know that many of you have been in limbo, and trying to prepare for this year without knowing whether you will be in a virtual, hybrid, or in-person environment.  Here in the math department at the University of Minnesota, we’ve been preparing for online teaching since June, and some of us have been teaching in an online/hybrid environment for years.  We’d like to use the MathCEP platform to share our experiences and resources with you.  This blog post is just the beginning. In this post, two postdocs at the University of Minnesota, Alexis Johnso...