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• Practical Activities for teaching Decision Mathematics

MEI Conference 29th July 2009

Jeff Trim - jefftrim@fmnetwork.org.uk

Graph Theory Word Maze

A puzzle activity for reinforcing Graph Theory vocabulary A Handful of Mathematicians

Short biographies of 5 pioneers of Decision Mathematics Graph Theory Vocabulary Hexagon Puzzle

Tarsia puzzle for revising definitions of common Graph Theory terms Instant Insanity

This activity demonstrates the power of Graph Theory to solve problems Bin Packing Exercise

A practical activity which practises bin-packing but demonstrates that the algorithms do not necessarily lead to an optimal solution

Algorithms for Sorting

The differences between four sorting algorithms are emphasised by using numbered playing cards and a practical approach

Sprouts

A pen-and-paper game using nodes/vertices and arcs/edges which can be easily analysed using familiar concepts

• There are 20 words associated with Graph Theory hidden in the square below. The Start and Finish squares are indicated and the words lie in a continuous path from one to the other. All moves are horizontal or vertical (not diagonal.). The initial letter of each word and the order in which they appear are given to stop you getting lost!

Start A R L A N E B I P A R E C P C Y C T O R E T D G E L E E N U T T I D E E R T C A W E G R E A I D J A L K D I A G R L A L P M O C H P R T P O E T E I M P L E E L O R E V S A H E N N X E T O D E M N I E O C E U N N A I O A C T E D L E R I L T N Finish

1. A

2. E

3. P

4. C

5. T

6. D

7. L

8. T

9. A

10. B

11. R

12. W

13. D

14. C

15. V

16. C

17. E

18. N

19. S

20. H

• Start A R L A N E B I P A R E C P C Y C T O R E T D G E L E E N U T T I D E E R T C A W E G R E A I D J A L K D I A G R L A L P M O C H P R T P O E T E I M P L E E L O R E V S A H E N N X E T O D E M N I E O C E U N N A I O A C T E D L E R I L T N Finish

1. A R C

2. E D G E

3. P L A N E

4. C Y C L E

5. T R E E

6. D E G R E E

7. L O O P

8. T R A I L

9. A D J A C E N T

10. B I P A R T I T E

11. R O U T E

12. W A L K

13. D I G R A P H

14. C O M P L E T E

15. V E R T E X

16. C O N N E C T E D

17. E U L E R I A N

18. N O D E

19. S I M P L E

20. H A M I L T O N I A N

• A HANDFUL OF MATHEMATICIANS Short biographies of five of the mathematicians who have contributed to

Decision Mathematics and Graph Theory

Leonhard Euler

Prolific 18th Century mathematician and founder of modern graph theory, after whom is named the Eulerian Cycle

Sir William Rowan Hamilton

Childhood mathematical prodigy and later Astronomer Royal of Ireland, after whom is named the Hamiltonian Cycle

Robert Prim

Developer of Prims Algorithm for finding a minimum spanning tree Joseph Kruskal

Inventor of Kruskals Algorithm for finding a minimum spanning tree Edsger Dijkstra

Computer scientist best known for the Dijkstra Algorithm to find a shortest path between two vertices

• EULER Leonhard Euler (pronounced Oyler) (1707-1783) b. Switzerland; lived and worked in Russia and Germany

Euler was an extremely prolific mathematician, contributing to the understanding of such diverse areas as calculus, topology, optics and astronomy. He popularised the use of the specific symbols i (for -1), e and . In particular, whilst the base for the natural logarithm had been discussed earlier by mathematicians with the symbol b, it was Euler who first used e to represent this constant in his book Mechanica of 1736. Although e is sometimes referred to as Eulers Number, his innate modesty precludes the thought that the letter was chosen because it was his initial! His name is given to the Eulerian Cycle; a closed path which traverses every edge in a graph exactly once. A graph which contains such a path is called an Eulerian Graph and the simple test is that the graph must be connected and have no vertices of odd degree. If there are two odd nodes, the graph will be semi-Eulerian, where the path starts and finishes at different points and so cannot be a cycle. The concept of an Eulerian Cycle relates back to a puzzle that Euler proved could not be solved concerning the seven bridges of Knigsberg. A feature of this Prussian city (now called Kaliningrad) was the seven bridges across the river Pregel and its tributaries. The challenge was to find a route whereby a citizen could cross every bridge once and return home again. In creating the mathematics to show that such a cycle was impossible, Euler became the founding father of topology. Euler lost the sight in his right eye after a life-threatening fever, although he blamed the loss on over-work. It is at this stage in his life that the well-known portrait (seen above) was painted. The sight in his left eye was subsequently lost due to a cataract. The fact that he continued to work, using his two sons as amanuenses, is testimony to his photographic memory and phenomenal skills of mental calculation. Amongst a range of other results named after him, is Eulers Identity: ei + 1 = 0. This remarkable brief statement combines the five most important mathematical numbers using the four basic operations of addition, multiplication, exponentiation, and equality, each used exactly once.

• HAMILTON Sir William Rowan Hamilton (1805-1865) b. Dublin, Ireland; lived in Ireland

William Hamilton was the son of a Dublin solicitor and his intellectually gifted wife, although both parents had died by the time William was 14. His prodigious talent was evident very early on and from the age of 3 he was sent to live with his uncle, Reverend James Hamilton in the village of Trim (about 20 miles from Dublin). This uncle, an accomplished linguist and polymath, instructed William, who at the age of 13 could claim that he had mastered a language for each year he had lived. Even prior to his graduation, at age 22 he was appointed Professor of Astronomy at Trinity College, Dublin, Director of the Dunsink Observatory and Astronomer Royal for Ireland. He was knighted at 30 and in 1837 became President of the Royal Irish Academy. He is considered to be Irelands greatest man of Science. Hamilton made valuable contributions to various different fields of science; algebra, optics, dynamics and quantum mechanics. Of interest to pure mathematicians is his pioneering work in the discovery of Quaternions, involving the extension of the concept of complex numbers into four dimensions. He may also be considered as the inventor of the cross product and dot product of vector algebra. In graph theory, a cycle which visits every vertex in a graph exactly once is known as a Hamiltonian Cycle. The still-unsolved Travelling Salesman Problem is a variation on this principle, where a tour of minimum length must be found which visits each vertex at least once. In 1857, in his spare time Hamilton invented The Icosian Game, based on the twenty vertices of an Icosahedron. The simple nature of the game was that the player, given the first five vertices, had to complete the route through the remaining fifteen vertices without repeats. The game was subsequently sold to a London games dealer, John Jaques & Sons, who marketed it in two forms flat for parlour use and hand-held for journeys under the name Around the World. The game was not a commercial success and Hamilton, who was paid 25 outright, had the better part of the bargain!

• PRIM Robert Clay Prim (1921- ) b. Sweetwater, Texas, USA

Robert Prim was a graduate of Princeton University, where he studied Electrical Engineering and subsequently (after World War II) also Mathematics, also remaining for a couple of years as a research associate. During the war years, he had worked as an engineer and been employed by the United States Naval Ordnance Laboratory. Robert Prim spent the late 1950s and early 1960s working at Bell (Telephone) Laboratories and then moved on to become Vice President of Research at Sandia National Laboratories. It was at Bell Laboratories, where he was Director of Mathematics research, that he developed Prims Algorithm for finding a Minimum Spanning Tree. It is for this that he is best known in Decision Mathematics. Together with his co-worker, Joseph Kruskal, two different methods were developed. In fact Prims algorithm had previously been discovered in 1930 by Vojtech Jarnik, was independently found by Prim in 1957 and subsequently rediscovered by Edsger Dijkstra in 1959. It is sometimes therefore called the DJP Algorithm! In contrast to Kruskals Algorithm, which focuses on the systematic selection of edges, Prims Algorithm builds up a set of connected vertices by progressively adding in the vertex least distant from the existing set, whilst ensuring none are revisited, thus avoiding cycles. The edges used form the minimum spanning tree of the graph. This algorithm was first published in the Bell System Technical Journal in 1957.

• KRUSKAL Joseph Bernard Kruskal (1928- ) b.

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