Day 3, MAT115

• some didn't do their readings,
• or got confused between the video and the reading "From Fish to Infinity".
• or confused the muppets (Ingrid was the one taking the order to the kitchen; tip-off in question 1! Ernie taught them to count...).

Let's review those definitions:

• Definition: a one-to-one correspondence is an association between two sets, so that each member of one set has a unique partner in the other set (and vice versa).
• Definition: a graph is a collection of points ("vertices") as well as a collection of arcs ("edges" -- each of which joins two points).

  An edge can even join a vertex to itself, which is called a loop in a graph.

  

• Definition: If every vertex in a graph is connected to every other (different) vertex by a single arc, we call it a complete graph.
• Trees are examples of graphs. Our trees so far have points which are numbers, and the arcs represent that one number is a factor of another number: Definition: A tree is a graph with two properties that graphs don't possess in general:
  • a tree has a special point (vertex), called "root";
  • a tree has no "circuits" -- no paths that lead off from a point and then arrive back eventually, as in the tetrahedron.
  That means that there's a unique path between any two vertices (which is not true for tetrahedron, because there are circuits in that graph).

Each time we add a new point (vertex), we have to connect it to the other points (vertices): so how do the number of connections (arcs) grow with the number of vertices (points)? We came up with a formula:

\[ \text{arcs}(n\ \text{vertices}) = \frac{n(n-1)}{2} \]

We realized that this is the sum of the first \(n-1\) natural numbers

Let's revisit that, with \(K_5\):

• the first vertex must connect to the other 4 (=5-1) vertices;
• each successive vertex then connects to one fewer (because the connections to the preceding vertices have already been made: 4+3+2+1=10.

I hope that you noticed the pretty geometric way of envisioning this in "Rock Groups", where the sum is the natural numbers from 1 to 10 (55):

I want to think back to the number 6, and beer:

• From the Washington Post: Our true feelings about race and identity are revealed in six words
• Me: Why six? I came up with "Mixing races makes life more fun!"
• Here at NKU one of our sociologists created a project called Mourning the Creation of Racial Categories -- six words!

Numbers (like squares), which can be represented by some geometric figure are called polygonal numbers. Another interesting example given in Rock Groups, are the perfect squares: \(n^2\) can be thought of as the sum of the first \(n\) odd natural numbers:

\(1+3+5+7+9\)

\(=25\)

These mysterious properties are parts of the personalities of numbers. What's your favorite natural number's personality? If your favorite number is 36, then you're lucky: it's both square and triangular!

It turns out that The Great Fraudini can read anyone's mind -- so long as they're thinking of a number between 1 and 63, and they're willing to play a little card game with him. So let's see how this works....

• Rules of the game: you show your "victim" every card. If they see their number on it, they say "Yes"; otherwise they say "No". Then you tell them their number!
• Can you become as powerful as The Great Fraudini? (You should be able to do it every time!)
• Here's the handout for the cards. You need a set of cards to play!

So in case you don't have a magic trick up your sleeve, I'm giving you one!

However I want you to think about it before I just give it away. Take a look at your cards, and see if you can figure out how The Great Fraudini is doing it....

But I also always tell my students that gambling is foolish, unless you have inside information. So I'm all about giving you the inside information. But generally, and (in particular):

But how might "primitive" people have counted? (Ernie's not primitive enough; neither is he a person.) You won't be surprised to learn that we will be using a tree to help us.

• Tallies
• using body parts -- "one hand" of sheep, say -- meaning five sheep.
• knotted strings
• cairns
  

• Let's suppose you need to let the King know how many sheep you have (but you were never taught how to count). You do, however, know how to make one-to-one correspondences:
  1. divide your sheep equally ("one for you, one for me") into two pens: either there is one sheep left over, or not; either they "correspond", or not.

     You make a note of whether there is a leftover sheep or not -- maybe you make a mark, like a "1" or a "0". This is all you have to do to communicate the number to the King!

     But you must also pay attention to the direction in which you write the marks.

  2. Send all the sheep in pen two (and any "left over") out to pasture, and then
  3. You divide the sheep in pen one into pens one and two: i.e., just do it again! And again, and again, and.... until you get down to a pen with just one sheep in it.
  4. Now let's think about how we might record the results to send to the King.

• The easiest way to illustrate the counting method is via a tree.

  Let's see how we might use a tree to represent the solution to the "22" counting problem: we'll use a ternary tree (three children).

  

  I say "Whoever's doing this..." -- I mean, "Whoever's doing this primitive counting...." This is presumably someone who doesn't know how to count -- at least not the way we do -- but they can tell if there's one left over (and write "1") or not (and write "0").

  Notice that we eventually have a single sheep in a pen, and that's when we're done. We have to write a "1" for the final sheep, to indicate that there's "one leftover".

  Then the answer will be written as 1, 0, 1, 1, 0. That is, from the bottom up, left to right. This is important! We have to have a consistent scheme for writing (or otherwise recording our results -- perhaps on a knotted string).

  So how do we write

  • 9 sheep
  • 31 sheep
  • 54 sheep

• Can you go backwards? How many sheep is meant by
  • 1,0,1,1
  • 1,1,0,1,1,0,0
  • 1,0,0,1,0,1,1,0,1

  (Notice that, while there can be either a 0 or 1 at the right end, there is always the "1" on the left -- meaning the last sheep standing!

  Can you rebuild the tree using these "tally sticks"? If so, you can get a job in the King's counting house.

• If you're afraid of sheep, think about counting pennies. When I was little I counted lots of pennies (I collected them, and would get rolls from banks to look through -- when I was done I had to put 50 pennies back into the rolls).

  But rather than count out fifty pennies, I'd make a stack of 10, then four more stacks of exactly the same height (one-to-one correspondence). Five stacks of 10 makes 50. This is a similar idea....

  If you have a lot of pennies, you could divide them in half, count half, and then multiply by two. But if you have one left over, you'd have to add that one.

  If the half is still too many to count, do it again (on half), and so on -- until you get to the point where you can count "the half" (like if it gets down to 1 coin, say!:).