Incredibly difficult math puzzle

For math/puzzle enthusiasts (as well for as my own future reference): This was one of the most diabolically difficult puzzles that I’ve ever seen. The object: use the numbers 1-9 exactly once in each row and column while ensuring that the given arithmetical operation in each cage is also correct. Here it is. Fair warning: while most MathDoku+ puzzles take me 20-40 minutes to solve, this one took me over 3 hours (spread out over 5 days).

Solutions to Exercises in Math Textbooks

I read a very thought-provoking blog post on the pros and cons of having answers in the back of math textbooks. The article and comments on the article are worth reading.

https://blogs.ams.org/bookends/2017/10/11/solutions-to-exercises-in-math-textbooks/

5 Ways to go Beyond Recitation

Most students will encounter recitation in a math class during their academic career. How can math professors make the experience more meaningful? MAA Teaching Tidbits blog has 5 ways educators can enhance the student experience during recitation.

1. Focus on getting students to do the work instead of doing it for them.
2. Incorporate group work into your sessions.
3. Get students to communicate what they understand to each other and to the class.
4. Have students relate mathematics to their own experiences.
5. Cultivate an environment where failure is ok and experimentation is encouraged.

My Favorite One-Liners: Part 114

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.

I’ll use today’s one-liner whena step that’s usually necessary in a calculation isn’t needed for a particular example. For example, consider the following problem from probability:

Let $X$ be uniformly distributed on $\{-1,0,1\}$. Find $\hbox{Cov}(X,X^2)$.

The first step is to write $\hbox{Cov}(X,X^2) = E(X \cdot X^2) - E(X) E(X^2) = E(X^3) - E(X) E(X^2)$. Then we start computing the expectations. To begin,

$E(X) = (-1) \cdot \displaystyle \frac{1}{3} + 0 \cdot \displaystyle \frac{1}{3} + 1 \cdot \displaystyle \frac{1}{3} = 0$.

Ordinarily, the next step would be computing $E(X^2)$. However, this computation is unnecessary since $E(X^2)$ will be multiplied by $E(X)$, which we just showed was equal to $0$. While I might calculate $E(X^2)$ if I thought my class needed the extra practice with computing expectations, the answer will not ultimately affect the final answer. Hence my one-liner:

To paraphrase the great philosopher The Rock, it doesn’t matter what $E(X^2)$ is.

P.S. This example illustrates that the covariance of two dependent random variables ($X$ and $X^2$) can be zero. If two random variables are independent, then the covariance must be zero. But the reverse implication is false.

My Favorite One-Liners: Part 113

I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.

My Favorite One-Liners: Part 112

This was also the story of my childhood.

My Favorite One-Liners: Part 111

I tried a new wise-crack in class recently, and it was a rousing success. My math majors had trouble recalling basic facts about tests for convergent and divergent series, and so I projected onto the front screen the Official Repository of all Knowledge (www.google.com) and searched for “divergent series” to “help” them recall their prior knowledge.

Worked like a charm.

My Favorite One-Liners: Part 110

I overheard the following terrific one-liner recently. A teacher was about to begin a lecture on exponential growth. His opening question to engage his students: “What does your bank account have to do with bacteria… other than they both might be really tiny?”

My Favorite One-Liners: Part 109

I tried a new joke in class recently; it worked gloriously.

I wrote on the board a mathematical conjecture that has yet to be proven or disproven. To emphasize that nobody knows the answer yet despite centuries of effort, I told the class, “If you figure this out, call me and call me collect,” writing my office phone number on the board.

To complete the joke, I said, “Yeah, this is crazy. So here’s my number…”

I thoroughly enjoyed my students’ coruscating groans before I could complete the punch line.