Session 3 Divisibility by 11 Problem

Here is my scratch paper from this summer as I worked on this problem while lounging on the grass pool-side.  It blew in the wind when I got in the pool and made a topologically interesting object!

I think I've got the answer, but I don't have a convincing proof yet that my result is the smallest possible.  Can you find a way to support your answer?

Math and Games Images

Konigsberg Bridge Problem  > Graph Theory > Topology

Atomic Structure  > Knot Theory > Science Applications

Kelvin's model of atoms as knotted ether is not pictured above but was being considered in the 1860s, so it would fall between Dalton's model and the plum pudding model.

Math as Game

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. 
A Mathematician's Apology by G. H. Hardy (London 1941).

Graph Theory Application

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Tutor Seminar January 17, 2013

A) Check-in

B) Handouts

C) Standard Techniques

D) Problem Solving

1) Schemper's Hardware has a number of bicycles and tricycles for sale.  If there are a total of 111 wheels and 46 seats, how many bicycles and tricycles does Schemper's have in stock?

2) Show that at a party of 20 people there are at least two people with the same number of friends at the party.  Presume that friendship is a mutual connection between two people.  (Hint: Consider the following three cases: i) Everyone has at least one friend at the party, ii) precisely one person has no friends at the party, iii) at least two people have no friends at the party.)

3) In any collection of seven natural numbers, show that there must be two whose sum or difference is divisible by 10.

Fractions and Fractals

Devaney's Madelbrot Bulb Site

http://www.ibiblio.org/e-notes/MSet/Period.htm