Instant Insanity

Although one might take the title of this particular blog post to be an assessment of the craziness that is the end of another semester, I assure it's not meant to be that way.  Quite the opposite in fact.  My semester went quite well, especially the Mathematics of Games and Gambling class that I designed and taught for the first time.  As I've down throughout the semester (though admittedly not nearly as often as I would have liked), it's time for another update for the course (here's a link to my last few course updates...published back in September and another in October).

Throughout the semester, I covered a variety of topics in the class (as a reminder, the class was designed for non-mathematics majors to take as an elective to fulfill distribution requirements).  The topics that covered included (but are not limited to):

• Chuck-a-Luck
• Roulette
• Craps
• Keno
• Five Card Stud
• Texas Holdem
• Lotteries
• Instant Insanity

It is the final topic that shall be the focus of today's entry.  

For a few semesters now, I've lobbied my mathematics department to allow me to teach an experimental course.  For the longest time, I wanted to do a 300-level Introduction to Graph Theory course.  Alas, our mathematics major is quite small (and most of our majors aim to teach at secondary schools so their electives get filled by education required classes like geometry).  As such, it has been deemed unlikely that I could ever get enough students to adequately fill up a Graph Theory course.

Last spring, I changed gears and proposed the Mathematics of Games and Gambling course instead.  As you can guess, that was approved (and has, for the record, been approved for the upcoming spring semester as well).  Although I couldn't teach graph theory, I still managed to find a way to squeeze in a few days of graph theory (at a basic level) by using the 1980s puzzle "Instant Insanity."

Before I describe how I taught the lesson, I will say that overall the lesson went extremely well.  I even put a question on the final exam which almost every student got correct!  If you have taught a lesson using Instant Insanity (or something similar), I'd love to hear about it.  If you happen to be inspired in some way to use what I did in your own classroom, I'd love to hear about your experiences when they happen!

Our class periods are just over an hour long, so for the first class of the unit I showed the students the original Instant Insanity advertisement after a short PowerPoint presentation that covered the various graph vocabulary that I wanted the students to know.  

The terms weren't difficult, but they weren't all simplistic either (bipartite graphs).  

The next class was much more of a riot (for me anyhow).  I gave each student a colored sheet Instant Insanity puzzle blocks which they had to cut-out and tape together.  Watching college students do something that many haven't done since a middle-school art class was definitely amusing (and somehow a bit worrisome too)!

After constructing the blocks, the students spent the remaining time trying to solve the puzzle.  While a few did succeed, the majority of the class did not successfully complete the confounding game.  I didn't provide any hints as they left class that day, though I did encourage them to find a "mathematical way to solve the puzzle."  

The final day of the lesson was the big one - how to solve Instant Insanity puzzles using graph theory. To begin, we discussed how each block could be "unfolded" to a two dimensional image.  

From there, we can easily pair the opposite sides.

Once the three pairs are made (front/back, top/bottom, left/right), all you need is a graph with four vertices (one for each of the four possible colors).  Draw an edge that for each of the three pairings.  For example, if the top color is blue and the bottom color is red, there should be an edge connecting the blue vertex to the red one.  If the left and right colors both happen to be yellow, there would be a loop connecting yellow to itself.  In total, there should be three edges for each graph (one graph per each of the four cubes).  

Using the four graphs (or if you prefer, a single graph that combines the four small graphs), you then look for a path that uses an edge from each cube once and that makes each of the four vertices have degree two.  Once you find one such solution, find a second solution.  The first solution corresponds to the front and back of the four cubes.  The second solution corresponds to the left and right.   Note:  Once you've found the front/back and left/right, the top/bottom is forced (and therefore you do not need to worry yourself about those)!

I've skipped over a few of the finer details, but in essence that's how you solve an Instant Insanity puzzle.  For those who want to give it a go, here are three puzzles.  It should be noted that some puzzles may not be solvable.

Good luck!

Letters of Recommendation...

Although I love almost all aspects of my job, there are a few things that I don’t necessarily look forward to doing.  One such thing is writing letters of recommendation.  Around this time of year, many of our undergraduate seniors are preparing for graduate school – and part of that preparation requires them to secure X numbers of letters of recommendation. 

Don’t get me wrong, I love (most) of our seniors – and I definitely want to write them great letters.  I truly believe that they will do well in whatever program they choose, none of that is the issue at hand here.  No, the issue is all on me.  I’m a harsh critic of myself, and while you probably can’t tell based on the drivel that I put on this blog (among the two blogs that I maintain), I do care.

A lot. 

Letters of recommendation are one of those things that can be super easy to write for a select student or two each year, but for the rest of the students…  Oh boy.  They can be exceedingly difficult, especially for students who I only had for a class or two or perhaps for a student who didn't get the best grades and/or perhaps didn't stand out in some other spectacular way.

For those of you who write letters for your students (or anyone for that matter), how do you do it?  How do you make the letters unique (or don’t you worry about that)?  Personally, I think a truly personalized letter is better – but then again, I've never served on an admissions committee.  I can’t say that I know for sure what they look for – and if I did, well, I’d guess my own letter writing skills would be a bit better!

The Price is Right LIVE: A Review

As I mentioned a few days ago, I took my Mathematics of Games and Gambling class to see a showing of The Price is Right LIVE at our local community arts center.  The show was last night and while I haven't seen my class yet to get their impressions of the show, here is my personal review of the Price is Right traveling show.

In a word:  Semi-lame.

Ok, that's not even a real word, but it's how I feel.

The good:

Many of people's favorite games are present.  Punch-a-Bunch, Plinko, Cliffhangers, Any Number, and Hole in One were all played.  The had one group of three people spin the big wheel.  Two people got the chance to play for the Showcase (only one showcase - both players bid at the same time).

The bad:

The show started late and ended early (or so it seemed).  Extremely short amount of time spent actually playing any games.  They showed a bunch of video clips from old shows - but nothing that you can't find on YouTube (for free).  The prizes were borderline good for most of the show...certainly not great (even with lowered expectations).  There was a refrigerator as a prize and a billiard table used as big prizes (neither were actually given away).  The small (initial bid) prizes included a 4 handheld phone system for a house (seriously, who uses those any more?), a pair of diamond earrings, a popcorn machine, and a vacuum.

The terrible:

The final showcase consisted of:
A new car (Nissan Versa I believe?)
A 3-day cruise to the Bahamas
An iPod touch
A 50 inch flat screen tv

The first contestant bid $19,000 and change.  The second contestant bid $20,000 and change.

For the traveling show, the person who is closest to the actual price (without going over) wins ONE of the items in the showcase - in this case, the 3-day cruise.  In order to win ALL of the items, you had to be within $100 of the actual price (i.e. not going to happen)!

Actual retail price (according to the show)?  $14,000 and change.

The show ended on that note.  A seemingly bogus final showcase, a pair of losing contestants, and a bunch of audience members feeling like the contestants were cheating.  The best line I overheard while leaving the theater:  "If a new car really cost $14,000, then everyone would have a new car."

Never mind all the other stuff in the showcase...

Since I was curious, here's a few numbers that I found via internet research*:
*note, all prices are guesses, I have no idea what the brands/companies were for some of the prizes
Price of 2012 Nissan Versa starting at:  $10,999 (from Nissan's website)
3-day Cruise:  $299
50 inch TV:  LG ($699.00)  one of the cheapest options
iPod touch:  $179.00
Total:  $12,176

So is the game rigged?  Well, I say yes but only because they make you think the showcase prizes are great when in reality they aren't nearly so good.  I also found it weird that in Punch-a-Bunch, the host knew exactly where the one $2500 prize was hiding...and it appeared to be printed on a larger card.  Makes me wonder if that particular hole had two cards residing in it, a $50 or similar prize if the contestant happened to select it and the big prize otherwise (the host showed the big prize to "prove" the fairness of the game).  I say when you have to "prove" that you are on the up-and-up, you probably aren't really on the up-and-up.

My suggestion to anyone who might see one of the live shows in their area - bid $1 and nothing more on the final showcase.  Chances are, your opponent will over bid and you then you win the cruise.  Don't bother trying to get too close, it won't work!

What I don't understand is why the producers of the show don't want to have one person win the cruise.  If the price is really only $300, that's paid for in a matter of 10 balcony tickets...a mere drop in the bucket.  Why have people leave angry (even if the pricing seems fair now that I looked up all the costs)?  If one of the two people had won the cruise, I think the entire audience mood at the end of the show would have been much better.

I also question the length of the show.  In a typical TV episode (granted, I'm sure footage is cut in order to fit it in 60 minutes - with commercials), there are six games played, the big wheel is spun by two groups of three people, and there are two final showcases.

In the travel show, there were only five games played, the big wheel was spun by ONE trio of contestants, and there was only one final showcase.  The entire show lasted just over an hour - and much of that time was "wasted" by showing the aforementioned video footage of old shows and for people making their way to the stage.

In the end, I'm interested to hear my students' take on the show, but for me, I can't in good conscience recommend anyone go to the show.  You'll have more fun watching old clips on your computer - save the price of the ticket.

The Price Really Is Right!

I've spent a fair bit of time discussing the happenings (and future plans) for my course that I am both developing and teaching this semester.  Today, I figured I would discuss our recent activity mostly to serve as another diary type entry for myself next semester.

My school's fall break is this week which means we get Friday off.  Yeah, a one day break isn't overly impressive (but we do get the Wednesday before Thanksgiving off as well which I appreciate).  Anyhow, it's often tough to cover much material in a short week for a distribution class since the students have all of Thursday, Friday, Saturday, and Sunday to forget.  However, thanks to our local community arts center, I got lucky.

You see, the traveling game show The Price is Right LIVE is coming to our town on Sunday.  When I learned about the show, I instantly arranged for tickets (free for the entire class) and rearranged my course schedule to accommodate the game show.  That meant both Monday and Wednesday were used to discuss Price is Right games and strategies.

Image source: http://content.clearchannel.com/cc-common/mlib/2036/01/2036_1264503009.jpg

We spent one and a half days discussing the mathematics behind the game of Plinko!  It ended up being a great way to review choice trees (using a small version of the Plinko! board).  It also natural evolved into a way to discuss using a small problem to help us solve a larger version...and it perfectly illustrated the need for a formula or shortcut when trying to analyze the "real" board.  In our case, the short cut is of course Pascal's triangle...which in itself is an excellent lead in to the next segment of the course where we will discuss the Binomial Theorem (and the MLB World Series).

After spending the better part of two classes on Plinko!, I used the remaining class time to give a crash course on some other games that might be played during the live show (I emailed the producers of the program but they wouldn't divulge which games would be played).

We looked at when to spin and when to "hold" during the Showcase Showdown.

We talked about bidding strategies.

We talked about basic strategies for a few of the games...and we ended yesterday's class with a video of what not to do which drew the expected laughs.

Overall, the students seemed quite excited for the opportunity to attend the show.  The math wasn't easy for the Plinko! board but most students seemed to intently study the various problems and explanations.  I had plenty of great questions about it during the lessons which convinced me that the students were truly engaging the material.  I believe the allure of winning a car (supposedly they play a game where someone could win a car even in the traveling show) was a great motivator!

Now that the lessons are over, I can only sit back and hope at least one of my students get the opportunity to go the stage.  It'd be a lot of fun to say that students can take my class and possibly win a vacation to Bermuda or something!

The Mathematics of Games and Gambling: A Course Update

I haven't done well keeping the blog updated on a daily basis (or even a semi-daily basis) but that doesn't mean that I haven't been keeping track of what has (and hasn't) worked in my classroom.  I've talked about my Mathematics of Games and Gambling course that I'm developing/teaching during the Fall 2012 semester a few times already (most notably here and here).  Since I haven't mentioned much about the course since then, it's probably time to reflect on how things have gone through (almost) five weeks.

So far in class (through five weeks - aka fifteen 65 minute classes), we have:

• Learned the game of Chuck-a-Luck
• Learned the game of Roulette (both American and European)
• Learned the game of Craps (both street and casino)
• Learned the game of Five Card Stud Poker
• Learned about expected values
• Learned about probabilities
• Learned about choice trees
• Learned about combinations and permutations
• plus a variety of other (smaller) topics such as a few brain teasers and card puzzles, plus assorted vocabulary as it arises in various gaming contexts.

We've also had a day where the class played craps, another where they played roulette, and today they happen to have a Five Card Stud Poker tournament.  The class has had five take home assignments so far (all but one spanning multiple classes).  Finally, the class took their first test on Monday of this week.

With all that's been done so far - and, quite honestly, that is a LOT of material for a distribution math class, it's time to reflect on the good, the bad, and the things that I might (or might not) change next time I teach the course.
*Note:  All observations are mine and mine alone.  I have not polled my students yet nor issued any sort of survey thus far in the semester.

Image source: http://singlemindedwomen.com/women-relationships/duchess-digest-the-good-the-bad-and-the-solution/

The Good:
Students love games and they seem to be working to understand the mathematics as a way to get better at the games.  I split the class into two separate tables when playing Craps - one table had every student (except one) turn their $300 into $1000+, while at the other table almost all the students went bankrupt.  It was a great, though unplanned, lesson in the draw of the casino!

The first round of testing had 6 of the 18 students earn an A on the exam.  For a distribution math course, that's awesome.  Even more impressive is the fact that the test was five pages long - and certainly not easy - and yet one-third of the class earned an A.

The best part of the class though isn't the grades or even the "fun" stuff.  Instead, the best part so far has been the students reaction to various problems, homework assignments, and in class activities.  The overall attention and engagement levels are through the roof as compared to a typical distribution class.  I can't say the class is at 100% in terms of engagement, but on any given day I would wager that at least 16 of the 18 students are fully engaged.  Again, a remarkable number considering the fact that most of the students in the class are self-professed math-phobes.

The Bad:
I mentioned the results of the first exam in "The Good" section, but I would be remiss if I didn't mention it in "The Bad" section as well.  Out of the 18 students, 4 of them ended up with Fs, including one student who left a pair of 21 point problems (out of 100 total exam points) blank.  As I said above, the engagement is there for most, but not all, students.

I think it's a bad thing that I have been unable to fit in time to play most of the games with the students and let them figure out their own strategies before we go over the mathematical analysis of each game.  I was able to do that for Roulette but not for Craps.

Things to think about changing:
I don't think I have the exact right balance of rigor and exploration.  I have a tendency to automatically analyze games logically as soon as I am introduced to them (my wife hates that, especially if we are playing against each other)!  However, just because that's a natural thought process for me, it most definitely is not for the majority of the students in the class.  I need to be better at guiding them through the analytical process.  In the beginning of the semester, I used a few brainteasers/puzzles as warm-ups which worked quite well.  Unfortunately, I haven't done much of that lately (mostly because the amount of material in the daily lessons hasn't allowed extra time).

I think the first exam should have been in week 4 rather than week 5.  I probably should have spent a full day on permutations and then a second day on combinations.  I ended up doing both on the same day and while it worked for 80% of the students, that's not close enough to 100% for my tastes.

Finally, I need to come up with a way to assess learning besides exams.  Many of the students in the class that didn't score great on the exam actually seem to know the material but they "froze" during the exam.  I think old math paranoia habits die hard...

I guess there's not really much purpose to this blog entry other than as a self-diary of sorts.  Of course, if you have any ideas for the course, by all means share them!