Day 12, Play-by-Play

Today we focused on some announcements first -- like your Canvas assignment due this Friday, your IMath assignment due on Monday, and the reading assignment for this Friday.
Then it was back to Yanghui's triangle (see the Day 12 page), where we talked about
1. the fact that we can probably figure out what 9 looks like. We can see the counting numbers, running down the diagonals after the diagonals of ones.
2. the existence of a 0 -- in fact, that their "bamboo counting rods" writing has 10 looking exactly like our 10, only on its side!
3. We figured out the error we'd identified in the triangle last time -- a 34, instead of a 35, was written by the creater of Yanghui's triangle. The scribe missed a stroke.
4. We noticed that there is place value: there is a tens place, and a ones place. It's used to make numbers from 10 up. We can presume that there's a 100s place, too -- we just didn't have any numbers that big.
5. We then used the back side of the sheet (the honeycomb pattern on the reverse side) to make a more modern version, called Pascal's triangle:
  
  If you click on that image, it will take you to a page that will tell you perhaps more than you want to know about Pascal's triangle:). But we'll know about all that stuff, by the time the course is over....
  Notice the asymmetry in their colors -- they could have run from left to right, just as well. The number patterns are the same, right and left.
  The first diagonal set of numbers they left out in the image above is called the tetrahedral numbers:
One of the things that Pascal used the triangle for was probability and statistical calculations. So we spent some time in class talking about the Hexstat probability generator, and how plinko works (see the Day 12 page for more on those).
Also the boardwork shows some of those connections. We connect Pascal's triangle up to binary trees, which we can think of as containing the results of tossing coins (Heads or Tails). Pascal's triangle shows, for each depth of the tree, how many different tosses lead to each type of result -- all heads, all heads but one, ..., all tails but one, all tails.
Then we talked about Facebook graphs. We think of these as very simple Facebooks, with just a few vertices (people) and edges (friendships). We looked at all possible ways of creating a distinct Facebook configurations, and discovered that there were

n # of possible edges # of Facebook configurations

1 0 1

2 1 2

3 3 4

4 6 11

(See the boardwork for today, and/or the Zoom video).
Furthermore, we saw that each configuration has a "dual" configuration, given by the rule that "If there's a connection (edge), break it; if there's no connection, make it!"
Next time we'll assign names to the vertices -- make them distinct individuals -- and suddenly we'll have a connection from these Facebook graphs to Pascal's triangle....

Website maintained by Andy Long. Comments appreciated.

n	# of possible edges	# of Facebook configurations
1	0	1
2	1	2
3	3	4
4	6	11