- You have an IMath assignment due soon, as well as another about Babylonian and Mayan mathematics. Pay attention to your schedule -- here, NOT ON CANVAS. Remember, Canvas is not your home page; this is. However, you will submit your written assignments on Canvas, as pdfs. Canvas is a tool, which we use; but our website is here.
- Last time we explored Mayan Math. The Mayans
- had a base 20 system -- they counted with their fingers and their toes!
- used "dashes and dots" to represent digits.
- did have one special place when doing solar/calendar
calculations: they changed the third place to 360
(=18*20), which we believe was "in honor" of the number
of days of the year (360).
- liked to group into four groups of fives -- perhaps one
for each appendage's extremeties.
- And rather than building sideways, like the Babylonians,
they built up! Toward the heavens....
- But perhaps the coolest thing about the Mayans was that they had a zero! They knew the value of nothing....
I like to teach you ways to make cold, hard cash, and this is the best one I've got. So learn this game! Play it for keeps, and you'll make a lot of money over your lifetime.
- Rules:
- This is a two-player game.
- An arbitrary number of counters is thrown down (e.g. toothpicks, pennies, M&Ms). The winner is the player who picks up the final counter.
- Player 1 must pick up at least one counter, but may not pick up all the counters.
- Then players alternate taking counters, with the rule that you have to take at least 1, but no more than twice what your opponent took on the previous play.
- To explain the game, I enlisted the help of Becky, one of your
fellow travellers on this journey through Math for Liberal
Arts.
Today she played a game or two of Nim against our friend The Great Fraudini; and, although she had a chance to win 20 dollars, alas! She did not take home the grand prize. However she did win the next three games! I invite you to watch their play.
- Becky and Fraudini talked about strategy, and how one might
approach determining how to play the game to win.
This game is truly an interesting game of chance: the Great Fraudini allowed Becky to choose the number of counters, and then to decide whether she would go first or second. This empowered Becky to assure herself of victory (but only if she knew the secret of the play).
I will tell you, however, that the strategy is just complicated enough that most people will never figure it out. Which means that, once you know it, you can take the lunch money off the other kids on the playground (or win a beer at the bar -- but only if you're of legal age, of course!).
- If you pay careful attention to the video, you will see that in
the first game -- which featured the $20 grand prize -- the
Great Fraudini makes a mistake (which Becky noticed later); he
could have won at a certain point, but he either
- genuinely screwed up, or
- generously allowed Becky a second chance to win the prize money! Because he was vulnerable when he made his move.
- Your job: figure out the strategy, with the aid of your
classmates on this
Canvas Discussion (please don't simply do
Google searches, etc. -- that kind of ruins the whole
exercise). And remember not to spill the beans if you get it
figured out! Provide hints, not answers.
- I'll give you one hint (which probably won't be very helpful): the
secret lies within Pascal's triangle!
- One thing you'll see emphasized within the play between Fraudini
and Becky:
A valuable strategy for the analysis of games like this is to consider simple cases: "Break a difficult problem into easier ones." Try to figure out a general strategy from those simple cases. Becky and I (oops, I mean "The Great Fraudini") considered several of those simple cases. Continue on with that process, and see what you might discover.