We've completed two Monday Math Madness contests. Last Friday Blinkdagger announced that Joshua Zucker, director at Julia Robinson Mathematics Festival, was randomly selected as the winner of the 2nd contest. Now it's my turn to post a contest problem. Those of you who are astute readers may have noticed that I said the contest would be held the 1st and 3rd Mondays of the month and today is actually the 5th Monday of March. Well, there's enough enthusiasm about this contest so we'll just do it every other week. So, we'll do 26 contests per year rather than 24. We're nice that way!
Earlier this month UC Berkeley professor emeritus of mathematics David Gale passed away. Gale made a number of significant contributions to mathematics and he loved puzzles, games, and finding beauty in mathematics. Gale's daughter had this to say:
On March 3rd Blinkdagger and I posted the first Monday Math Madness problem. On March 11th, after the first contest ended, I posted a couple of different solutions to the problem. Pat Ballew, even though he wasn't picked as the random winner, impressed me with a very clever solution to the problem that generalizes very nicely. He uses an approach called Markov state matrices, which I had never heard of. It seems to me that this approach is pretty similar to the one I posted from Richard Berlin. Pat and I exchanged several emails where he explained the method and here is my attempt to explain what Pat explained to me.
This was the problem:
A popular blog has just three categories: brilliant, insightful, and clever. Every blog post belongs to exactly one of the three categories and the category for each post is selected at random. What is the probability of reading at least one post from each category if a reader reads exactly five posts?
Pat's approach starts by creating a matrix that encodes the probabilities of going from one "state" to another as a new blog post is read. State just refers to whether 0, 1, 2, or 3 categories have been encountered after reading some number of blog posts. After one blog post has been read we are in state 1 (1 category has been read). After two posts have been read we may be in state 1 (if both blog posts are in the same category), or state 2 (if the two categories are different), but not in state 3 (you could not have encountered three categories after having read only two blog posts.)
Here's a joke I got a good chuckle out of. I'm not sure who to credit since there a number of web-sites with this joke so I'll credit the site where I first found it, Savage Research.
Two mathematicians were having dinner in a restaurant, arguing about the average mathematical knowledge of the American public. One mathematician claimed that this average was woefully inadequate, the other maintained that it was surprisingly high.
"I'll tell you what," said the cynic. "Ask that waitress a simple math question. If she gets it right, I'll pick up dinner. If not, you do." He then excused himself to visit the men's room, and the other called the waitress over.
"When my friend comes back," he told her, "I'm going to ask you a question, and I want you to respond `one-third x cubed.' There's twenty bucks in it for you." She agreed.
The cynic returned from the bathroom and called the waitress over. "The food was wonderful, thank you," the mathematician started. "Incidentally, do you know what the integral of x squared is?"
The waitress looked pensive; almost pained. She looked around the room, at her feet, made gurgling noises, and finally said, "Um, one-third x cubed?"
So the cynic paid the check. The waitress wheeled around, walked a few paces away, looked back at the two men, and muttered under her breath, "...plus a constant."
Here are some interesting recent Math-related posts in the "blathosphere."
You have only a couple more days to get your solutions in for the second Monday Math Madness at Blinkdagger.
Blinkdagger has posted the second Monday Math Madness contest. It has a fun St. Patrick's Day theme. Check it out!
You've got a week to solve this problem and send in your well-explained solution. The Blinkdagger guys are giving out $10 in Amazon gift certificate cash to a randomly-selected winner.
April 7th will be the next contest at Wild About Math! I'll be giving away something more fun that cash next time so check back here but solve the Blinkdagger St. Patty's Day problem first.
We interrupt your regularly scheduled blog for a non-Math post ...
It's Saturday night here, I was just memed by Robert at Reason-4-Smile, I don't feel like doing any productive work so I'll answer the meme and launch it further into the blogosphere.
The meme asks me to give a link to the person who meme'd me, to tell 7 interesting things about me, and to then pass the meme on. I interpreted the meme to mean that I should tell things that you're not likely to guess. Here goes:
Today is March 14th, or 3/14 as Americans write dates. Think that pi is roughly 3.14 and you'll see why today is Pi Day.
Denise at Let's Play Math does a superb job of writing about this special day. She's got a Pi poem, lots of links to pi-related pages and even a hokey but very funny mathematical pi song.
Math Mom is also commemorating this special day with some ideas about how to celebrate the day. Plus, there are other ideas in the comments of Math Mom's post.
I wanted to state what some of you may have been noticing: My rate of posting has decreased, posts have been more superficial, and I've been slow to respond to comments. Life has been busier than I expected, with important life matters and paid short-term project work filling a significant percent of my time. The slowness may last several weeks. I'm not dropping this blog as I enjoy the value that I give and receive through this community. I may not get to do a significant post until early next week. Stay tuned for the Blinkdagger contest post Monday, and for my next contest the first Monday of April. Also, do subscribe to my mailing list to get Math Bite emails with interesting little Math tidbits.
In upcoming posts I'll share a very nice alternative approach to the Monday Math Madness contest I received and I'll review an outstanding Geometry book for those of you who want a real work out when doing Geometry.
I'm also going to experiment with doing some short posts that tell you about what other Math bloggers are writing about to keep the momentum moving. And, if any of you would like to write some guest posts to bring some attention to your blogs I'm very open to that possibility.
For the very first Monday Math Madness contest we got 13 submissions. Of the 13, 6 were correct. For the record, I solved the problem by enumerating the various cases where 3 categories were represented and computing and adding their probabilities. I also verified my solution to the problem by writing a computer program to enumerate all 243 (3^5) permutations of 3 categories and 5 blog posts and count the ones were all 3 blog categories were represented. So, I'm pretty confident I got the right answer