Day 5, MAT115R

"If he succeeds in finding a pattern in prime numbers, he will hold the key to every computer in the world."

Definition: If every vertex in a graph is connected to every other (different) vertex by a single arc, we call it a complete graph.

Each number, considered as a complete graph, encompasses all the numbers that go before it:

Name of graph:
Number of arcs:	\(T_0=\frac{0(1)}{2}\)	\(T_1=\frac{1(2)}{2}\)	\(T_2=\frac{2(3)}{2}\)	\(T_3=\frac{3(4)}{2}\)

and we drew a few more (\(K_7\) and \(K_8\)).

• Today I teach you how to count! You're probably thinking "Heck, Sesame Street taught me how to count; Ernie taught me how to count last week!"

  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.

• We have some good ideas of how "primitive" people counted, generally via a one-to-one correspondence of some sort (hopefully you read the article I assigned for homework, Early Concepts of Number and Counting).
  • Tallies
  • using body parts -- "one hand" of sheep, say -- meaning five sheep.
  • knotted strings
  • cairns
    

• But today I want to describe a method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht described at the 2011 National Math Meetings (based on a suggestion from 1817!):
  
  • 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 branch off of a parent, rather than just two -- as we've seen in the binary trees for prime factorization).

    

    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!:).

Watch Dave Brubeck's leg, counting out the rhythm....