Travis Hoppe

Plus-size numbers They’re just big boned, more to love really

Venn-diagrams Forget everything you’ve forgotten about them from

third-grade

Football betting Did I mention I’m from Vegas?

First a contest! Write the largest number you can think of on the note

card. Use standard mathematical functions or define your

own. The number must be verifiably finite and computable. The number must be completely defined on the card.

First-grader:

Third-grader:

Sixth-grader:

Twelfth-grader:

9999999999

999999999

9999999

99

1000000000

Addition, multiplication and exponentiation are simply higher orders of the same function:

copies of b a

ab a a a

copies of

b

b a

a a a a

The idea is not to generate the largest number, per se, but rather the largest growing function...

Many different styles: Conway’s chained arrow, hyper-geometric, and of course Knuth’s up-arrow

Each arrow starting from exponentiation forms the higher operators:

copies of

b

b a

a b a a a a

...

copies of copies of

a

b a

b a b a

a b a a a a a

3

33

3

3 27

3 7625597484987

3 2 3 27

3 3 3 3 7625597484987

3 4 3 3

33 27

3 93

3 3 3 3

3 3 (3 ) 3

Note that the operator is right-associative:

Clearly we can grow larger numbers by simply adding more arrows onto the expression:

copies of b a

a b a a a

1 1 1

copies of

n n n n

b a

a b a a a

We have primitive brains – For small numbers we can only think spatially, four

cows, three hens etc … Abstract numerical systems allow us understand larger

quantities If you build it ….

Large numbers systems were invented because of their necessity. For example …

Graham’s number is so big that even Knuth’s up arrow notation is insufficient to contain it. It is the best known upper-bound to the problem:

Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2^n vertices. Then color each of the edges of this graph using only the colors red and black. What is the smallest value of n for which every possible such coloring must necessarily contain a single-colored complete sub-graph with 4 vertices which lie in a plane?

1 3 3g

13 3ngng

64G g

This is an upper bound to the problem. It has been proven that the lower bound solution is at least 11. The authors state that there is, “some room for improvement”.

You, at the conclusion of this talk

Let C = { C1, C2, ..., Cn } be a collection of simple closed curves drawn in the plane. The collection C is said to be an independent family if the region formed by the intersection of X1, X2, ..., Xn is nonempty, where each Xi is either int(Ci ) (the interior of Ci ) or is ext(Ci ) (the exterior of Ci ).

If, in addition, each such region is connected and there are only finitely many points of intersection between curves, then C is a Venn diagram, or an n-Venn diagram if we wish to emphasize the number of curves in the diagram.

In other words – every subset built from a collection of n objects has to be represented only once ….

Not Venn as A U B is not represented, thisis still known as an Euler diagram

Constructing graphs allows different Venn diagrams of the same order to be compared.Diagrams are isomorphic if their graphs areIsomorphic.

Number of vertices in the graph is no more than:

2 2

1

n

vn

Must display “n-fold” symmetry. Can be shown that these only exist when n is

prime

Football betting

0 1 2 3 4 5 6 7 8 90123456789

Which squares are better and by how much?

Should some squares cost more?

Are you more likely to be a win or lose?

Is the post-season different from regular season?

Data collection – wrote script to dump all scores from 1994-2007 season (no box scores were found pre-1994).

Partitioned data into regular season and post (wildcard, playoffs and Super Bowl) games.

Took into account home-field advantage; 07 is different from 70, with the first score the home team (irrelevant in games where neither team has home field advantage).

Each score (x,y) was given an expected value:

431 2( , )( , ) 10 02 ( , ) 3 ( , ) 4 ( , )0 0Q Q QQP x y P x y P x y PEX x y x y