Archive for the ‘Math 30-1’ Category

You may not know it, but the Amazing Race is big in Canada. It is so big, in fact, that they are planning on running a Canadian version. I’ve already started training and I am currently accepting applications from people who would like to join me on the winning team. But I digress…

Last week’s season finale (season 21, I think) included a challenge that asked contestants to put banners containing the words “hello” and “goodbye” beneath country flags, in that order, and in the language from that country. The contestants struggled and the challenge took over 2 hours, but one contestant tackled it systematically by trying all possible combinations. It was made for a math classroom. In the WNCP, this fits Permutations and Combinations from Pre-Calculus 12 (Math 30-1) in Alberta. It also fits Math 30-2 in Alberta. Here it is, in 3 Acts.

Act I

Play the video by clicking the photograph of one of the contestants working on the challenge.

Flag Challenge

With any kind of luck, the students will wonder how many combinations of the “hello” and “goodbye” banners are possible. They will require more information.

Act II

This video is longer than the Act I video, and by watching it closely, they should be able to determine that they are working with 9 country flags, and 20 banners with words on them. There are 2 extra banners.


I don’t have a video with the answer. It is fun playing with this problem, though. Initially, there are 1,216,451,004,088,320,000 combinations (20 x 19 x 18 x…x 3). By getting France and Spain correct immediately, the contestants reduced that number by a factor of 116 280, and now only have 10,461,394,944,000 possible combinations to try. If they had truly had to guess them all, they’d still be at it.

Enjoy. Fix my math.

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I’ve been getting a kick out of the first season of this show.

Act I

Just to pique their interest, play this clip.  Ask them what they wonder about. Hopefully they talk about the number of possible codes.

Act II

Scene 1

Play this clip. Let them work.

Scene 2

Play this clip. Let them work.


I have no video that reveals an answer here. Let them share their solutions with each other. Then let the watch the clip below so they can at least find out if Fusco manages to get the file.

I may have learned a new trick. It’s possible that this link will take you to a zip file (10.2 MB) that will allow you to download all 4 videos. It’s possible that it won’t. Let me know, either way.

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I got this idea from a teacher I marked diploma exams with a couple of years ago.  I’d love to credit him, but I can’t remember who he is.  This is a WCYTWT submission, or as I like to call it, WWDDWT (What Would Dan Do With This).  I used it with a couple of classes last year, and they really enjoyed it, but I used it poorly.  I did it at the end of a unit on Permutations and Combinations, so the students knew exactly what to do with it.  It wasn’t truly a problem, because they knew exactly how to solve it when I gave it to them.  I should have used it on day one of Permutations and Combinations, and let them invent the fundamental counting principle on their own.

Mozart’s Dice Game

Mozart is credited with creating a dice game, whereby you roll a pair of dice 16 times to select 16 measures to insert into a minuet section, and then a single die 16 times to select 16 measures to insert into a trio section.  For example, if you roll a 6 for the first measure, you consult a chart to see what measure number to insert into the first measure of your minuet, and so on.  The idea is that no matter what you roll, you always produce a minuet that fits whatever rules go along with a minuet.  This site explains it in a little better detail.  Mozart’s Dice Game.


You give the students the history, and then you throw this site up on the SMART board.   Play Mozart’s Dice Game has a chart that looks like this:

The drop down menus let the students enter numbers that they roll on dice, so give the kids some dice, and let them enter their rolls.

Notice the link below the chart that says “Make Some Music!” Once the students have all entered their rolls, you click here and the newly created minuet will play.  There’s even a “Generate Score” link below the media player that lets you generate the score for the minuet they created.  My students loved this, and printed theirs off to try to play it on the piano themselves.

After the minuet plays, you tell the kids that even though this was written more than 200 years ago, you are pretty sure that nobody has heard the particular minuet the class just created ever before.  Ask them to discuss this statement.  They’ll say things like, “Why, did they lose the score until recently?” and dance around it until one kid finally asks, “How many minuets could be made in this game?”  Then you’ve got them and you let them play around with it.  They will invent the fundamental counting principal, determine that there are an incredibly large number of possible minuets, and even create their own interesting extensions.

One student even came in the next day with an iPhone app that generates minuets using Mozart’s Dice Game.  The only  problem with the app is that it only randomly generates minuets, and doesn’t allow you to enter your own rolls.

Update: March 11, 2016

My greatest fear was realized. The site I talk about above is no longer functional. Something about Midi Files, Flash players, and the like. I couldn’t get it to work in any browser.

I found a functional site. https://www.mozart-game.cz/

It randomly selects the measures to play, as highlighted below. You can play your own measures by clicking on the un-selected ones, so you could still have your students roll dice and create their own minuet, but you’d have to click your way across the chart. If you can keep time, it’s pretty easy. The vertical columns represent one roll of a pair of dice. So if the first roll was a 2, you’d play measure 96. In the screen shot below, the first roll was 3, the second roll was 6, and so on.



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For your viewing pleasure, here are what I believe to be all the entries in the binomial expansion contest.  They are presented without bias, and in no particular order other than putting the best one at the end.


James Tanton’s Part 1

James Tanton’s Part 2

Jason Dyer’s


Edit (May 30):  Jason Dyer’s was judged the winner in the competition.  See here for the rationale for the decision.  Congratulations, Jason.

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