(May 26, 2016)  Split into 3 teams.  Each team gets 5 dice.  Roll them all, look at them, but don’t let the other team see them.  The first team makes a 2-digit bid.  The first digit of the bid is your prediction of the total number of dice of a certain value every team combined has.  The second digit is your prediction of what that certain value is.  So a bid of 53 means that you probably got at least 1 or 2 threes, so you’re prediction that once everyone shows their dice, there will be at least 5 threes on the table.  The next team bids something higher.  The third team then does.  But wait.  If you don’t want to raise the bid, you can call out “Dudo,” which means “I doubt.”  Then everyone has to show their dice.  If the most recent bid is correct, you lose a die.  If your doubt (“dudo”) was correct, the last person who bid loses one.

This is easier to understand from Gordon Hamilton’s video.*  But we didn’t play it quite the way he did.  Our kids decided to make the rules themselves.

The written rules that I’ve seen state that to raise a bid, you can increase one digit or the other digit or both.  The kids were adamant that as long as you raise one digit, even if you decrease the other, this is a raised bid.  I went against everything I believe (and try to practice) in terms of math pedagogy and tried to convince them to follow the conventional rules.  “What if you want to play this with others – it’s a famous game!” and “What if you want to answer the unsolved question about this?” etc.  Fortunately, the kids were adamant that they wanted to do it their way.

BTW the unsolved question about this game is “What is the best strategy for winning?”

BTW again, this question is a good way to introduce students to probability.



A few students who had been absent wanted to see the progress made on the sequence of Hailstone Numbers (those formed by the Collatz conjecture sequence – halve it if even, 3n+1 if odd).  Our group – particularly M – has been working on the total stopping time if you start with 888.  So calculations on that continued.   They were up to a total stopping time of 64 (64 moves/calculations!) by the time class ended – see the awesome photo of their work on this so far.



We had 10 minutes left in math circle for this school year.  I wanted the kids to leave this course with an awareness that the realm of mathematics goes way beyond number theory and graph theory.  Those two areas have been the most visited in our course.  So I offered them a chance to try an unsolved problem from the mathematics of origami (paper folding), which delves into graph theory (again!), combinatorics, and geometry (finally!).  While origami is considered an art, it has applications in engineering, materials science, biology, and more.

I asked an unsolved problem that Thomas Hull first posed in a 1994.  Gordon Hamilton explains it this way:  “Given creases on a piece of paper, how many ways can you fold it flat?”  (See Hamilton’s video “Fold it Flat” for a demonstration.**)  The section of Hull’s paper that deals with this is called “Problems with Extending Globally.”  In other words, we can crease a paper this way, and have x many ways of folding it flat.  We can crease the paper that way and have y ways of folding it flat.  And so on.  But what’s the rule that applies to all ways of creasing?  We’d like to be able to say that no matter how you crease it, there is a way to predict how many ways of folding it there are.

I let the kids choose whether to work on Collatz or origami.  Two chose origami, the rest Collatz.  As they worked, I worked with them to recap the unanswered problems we’ve done to be sure they each had something to think about over the summer, if they want to.



Thank you for sharing your kids with me during these wonderful 6 weeks.  This math circle has been unusual for me in that we worked on so many different problems, instead of stretching one out for 6 weeks.  I wanted to give kids opportunities to dwell on them at home, in bed, late at night.  Different problems capture different people’s fancies.  Ask your kids which problems intrigued them the most.  For me, it has been the No Three in a Line problem.  (This one seemed to interest the kids the least.  Oh well.)  I did that one in a math circle with 5-6 year olds years ago, and it didn’t catch my fancy then.  Now it has.

Hope to see you in math circle the fall!  We hope to have new schedules posted soon.





*https://en.wikipedia.org/wiki/Liar%27s_dice  (Hamilton calls the game by one of its original names, Perudo.  I used its name Liar’s Dice in class because a few of the kids already knew it by this name.)

**http://mathpickle.com/unsolved-k-12/ (Click on “Grade 5” to see “Fold it Flat.”)


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