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
E
D
D
E
G
R
E
N
E
C
R
C
G
E
A
R
T
E
N
O
T
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
L
P
E
E
I
L
P
L
X
C
E
A
C
L
R
D
A
O
O
E
E
D
N
Y
E
T
J
L
E
R
T
U
L
E
C
E
C
A
P
T
E
O
N
E
B
T
N
A
L
M
E
V
D
N
R
I
O
U
W
K
O
I
S
E
A
I
P
R
T
E
D
C
M
A
M
I
L
A
E
T
G
I
H
P
H
N
O
T
R
T
I
R
A
P
L
E
I
A
N
Finish

Start
A
E
D
D
E
G
R
E
N
E
C
R
C
G
E
A
R
T
E
N
O
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L
P
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A
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N
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J
L
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C
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T
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N
E
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
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
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
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
E
B
T
N
A
L
M
E
V
D
N
R
I
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W
K
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A
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D
I
A
N
Finish
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 Prim’s Algorithm for finding a minimum spanning tree
Joseph Kruskal
Inventor of Kruskal’s 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 π. 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 Euler’s 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 Kőnigsberg. 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 Euler’s Identity: eiπ + 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 Ireland’s 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 1950’s and early 1960’s 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 Prim’s 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 Prim’s 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 Kruskal’s Algorithm, which focuses on the systematic selection of
edges, Prim’s 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. New York, USA
Joseph Kruskal is a graduate of the University of Chicago with a PhD from Princeton
University. He is a fellow of the American Statistical Association and former
President of the Psychometric Society. He has also campaigned for civil rights and
fair housing.
Joseph Kruskal worked with Robert Prim at the Bell Laboratories where they
developed two different approaches to the same problem. Formerly know as the Bell
Telephone Laboratories, their practical application was focussed on communications
networks. These can be represented as graphs with weighted edges.
Kruskal’s Algorithm, also known as the ‘Greedy algorithm’, is a method for finding
a Minimum Spanning Tree for a given network by choosing in turn the edges with the
lowest weights whilst avoiding the creation of cycles. In fact the term greedy
algorithm is a more general expression meaning a process which chooses at a given
stage during each iteration the best option at that point. Kruskal’s Algorithm was first
published in the ‘American Mathematical Society Proceedings’ in 1956.
Kruskal later said that he regretted that the term minimum spanning tree has taken
hold, claiming that ‘minimum’ is too vague; minimum in what way? He stated: “I
always think of the concept as ‘shortest spanning subtree’ and hope someday to see
SST replace MST.”
Joseph Kruskal also had two brothers who were renowned in different areas of
mathematics and physics; Martin Kruskal, the co-inventor of ‘solitons’ and ‘surreal
numbers’ and William Kruskal, who developed the Kruskal-Wallis one-way analysis
of variance.
DIJKSTRA
Edsger Wybe Dijkstra (pronounced ‘Deekstrah’) (1930-2002)
b. Rotterdam, Netherlands; worked in Amsterdam and also Texas, USA
Edsger Dijkstra studied Theoretical Physics at the University of Leiden before
moving into the field of Computer Science. He worked for the Mathematisch
Centrum in Amsterdam and then the Burroughs Corporation, which had progressed
from making adding machines to typewriters and computers. He later held the
Schlumberger Centennial Chair in Computer Sciences at the University of Texas.
He wrote prolifically on programming and observed that “programming is so
inherently difficult and complex that programmers need to harness every trick and
abstraction possible in hopes of managing the complexity of it successfully”. He was
renowned for working carefully in fountain pen and then photocopying and
distributing these manuscripts, each numbered with EWD as the pre-fix. More than
1300 of these hand-written articles are known to exist.
In Graph Theory, he is known best for Dijkstra’s Algorithm for finding the Shortest
Path between two points in a network, which he developed in 1959. This is a
labelling procedure which identifies the smallest total path length to each vertex from
the starting point. Tracing back from the finish vertex finds the shortest path itself. It
is an example of a greedy algorithm because of the step during each iteration where
the vertex with the lowest working value is chosen as the best and is then permanently
labelled.
He received the 1972 Turing Award (after Alan Turing, the British mathematician
who helped break the German Enigma code in WWII) for fundamental contributions
in the area of programming languages.
Despite a career in programming, Dijkstra only owned one computer in his life. Even
then it was after pressure from his colleagues and just used for email and internet
browsing. This is in keeping with his quoted opinion that: “Computer Science is no
more about computers than astronomy is about telescopes.”
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Digraph
 
Variations of this puzzle have been published and marketed under various titles
over the years, but all have the same underlying design. The challenge is to
stack four cubes such that each side of the stack shows four different images.
The puzzle can be very frustrating to solve by trial and error, but yields very
quickly to the approach described below. It is an excellent way to introduce
applications of graph theory without straying into the common algorithms on the
syllabus such as minimum spanning trees and shortest paths. In particular, it
gives students a chance to use vocabulary such as vertex, edge, and sub-graph.
Preparation:
Nets for the puzzle can be found here:
http://nrich.maths.org/public/viewer.php?rss=1&obj_id=443&part=index
Symbols can be used for the cube faces instead of colours.
Duplicate the puzzle sheet onto four different pale shades of paper and roughly
cut out the nets. Distribute so that each student has each of A, B, C and D in a
different colour. Paper is ideal for this (rather than card) and glue is not
required. Scissors and coloured pencils to match the paper shades are needed.
It really benefits the finished cubes if the folds are creased firmly first!
Method:
When students have made up the cubes and tried the problem, introduce the
idea of using a graph to represent the symbols on opposite sides of each cube.
Prepare a graph with four vertices. Assign one symbol to each vertex.
For each cube, add three colour-coded edges to the graph. Each of these three
arcs will connect the vertices corresponding to the symbols on each pair of
opposite faces.
The resulting graph will contain twelve arcs, three of each colour.
Sub-graphs:
The students need to find and draw two distinct sub-graphs, each of which
contains one arc of each colour and in which each of the four vertices has
degree two (i.e. is a ‘two-node’). The sub-graph need not be connected, but I
have chosen two connected sub-graphs for my own solution.
One nice outcome is that students will have a graph specific to their
combinations of colours and therefore probably different to the person next to
them! However, if the vertices have been labelled in the same way, at least the
graphs will be isomorphic, ignoring the colours.
Interpreting the Solution:
Label one sub-graph “Left-Right” and the other “Front-Back”.
Label the clockwise direction by each graph.
Left – Right
Front - Back
The arcs of each colour, read clockwise, indicate how each cube is to be
arranged in the tower.
E.g. a green arc from flower to circle (as you read clockwise) on the Left-Right
sub-graph means that on the green cube, the pair of opposite faces that are the
flower and circle need to be arranged with the flower on the left and the circle
on the right (from the student’s point of view!).
It is often necessary to demonstrate how to do this for at least one cube!
The solution omits one arc from each cube used in the original graph. These
arcs correspond to the top and bottom faces that are concealed in the tower.
 
The solution shown here is based on the four cubes having the following colours:
A – Green
B – Purple
C – Orange
D – Blue
Graph:
Sub-graphs:
Left – Right
Front - Back
[See also p. 13-14, AEB Discrete Mathematics, Heinemann, 1992.]
  
Where objects of varying sizes must be placed into containers of a fixed
capacity, the problem is described as ‘bin packing’. The name is used for any
problem of this general type, whether to do with objects, lengths, times or
whatever the scenario.
Two algorithms are commonly used to attempt to solve these problems:
First-Fit Bin Packing:
Number the bins, then always place the next item in the lowest numbered
bin which can take that item.
First-Fit Decreasing Bin Packing:
Reorder the items into decreasing order of size.
Number the bins, then always place the next item in the lowest numbered
bin which can take that item.
Activity
A builder uses piping of standard length 12 metres.
The following sections of varying lengths are required for a particular job:
Section
Length
(metres)
A
B
C
D
E
F
G
H
I
J
K
L
2
2
3
3
3
3
4
4
4
6
7
7
Explore how many standard 12 m lengths of pipe will be required if each of the
following methods is used:
(a) First-fit bin packing
(b) first-fit decreasing bin packing
(c) trial and improvement
[Adapted from p. 209-210, AEB Discrete Mathematics, Heinemann, 1992]
Length of standard pipe section (metres)
Length of standard pipe section (metres)
Bins
12
11
10
9
8
7
6
5
4
3
2
1
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Pipe Sections
A(2)
B(2)
C(3)
E(3)
D(3)
K(7)
L(7)
K(7)
L(7)
F(3)
G(4)
J(6)
H(4)
I(4)
A(2)
B(2)
C(3)
E(3)
D(3)
F(3)
G(4)
J(6)
H(4)
I(4)

(a) First-fit bin packing
12
Length of standard pipe section (metres)
11
10
9
D(3)
G(4)
8
7
6
I(4)
C(3)
5
F(3)
4
3
B(2)
L(7)
5
6
7
8
6
7
8
H(4)
2
1
K(7)
J(6)
E(3)
A(2)
1
2
3
4
(b) First-fit decreasing bin packing
12
A(2)
Length of standard pipe section (metres)
11
F(3)
10
9
G(4)
H(4)
I(4)
8
E(3)
7
6
5
4
D(3)
K(7)
L(7)
J(6)
3
2
C(3)
B(2)
1
1
2
3
4
5
(c) Trial and improvement
12
Length of standard pipe section (metres)
11
A(2)
B(2)
F(3)
I(4)
10
9
C(3)
D(3)
8
E(3)
7
H(4)
6
5
4
K(7)
L(7)
J(6)
3
G(4)
2
1
1
2
3
4
5
6
7
8
ALGORITHMS FOR SORTING
BUBBLE SORT
Step 1:
Compare the first two numbers.
Step 2:
If the first number is larger than the second, exchange the numbers.
Step 3:
Repeat steps 1 and 2 for all pairs of numbers until you reach the end of the list.
Step 4:
Repeat steps 1 to 3 until no more exchanges are made.
SHUTTLE SORT
The Shuttle Sort works by comparing pairs of numbers and exchanging them if necessary.
Step 1:
Compare the first two numbers and exchange if necessary.
Step 2:
Compare the second and third numbers and exchange if necessary,
then compare the second and first numbers and exchange if necessary.
Step 3:
Compare the third and fourth numbers and exchange if necessary,
compare the second and third numbers and exchange if necessary,
compare the second and first numbers and exchange if necessary.
Step 4:
For a list of length n, continue until n-1 passes have been performed.
SHELL SORT
The shell Sort differs from the Bubble and Shuttle methods as it compares, and possibly
exchanges, non-adjacent elements.
The set of elements is split into subsets. The number of subsets for the first pass is INT(n/2),
that is, the number of elements, divided by two and ignoring any remainder.
QUICK SORT
Step 1:
Choose any number x from the list L,
(usually the number at the mid-point – but for AQA, choose the FIRST).
Step 2:
Write all the numbers smaller than x to the left of x, reading the original list from left to right.
These form a new list L1. Write all of the numbers larger than x to the right of x reading the
original list. These numbers form a new list L2.
Step 3:
Apply steps 1 and 2 to each separate list until all of the lists contain only one number.
Step 4:
The original list is now in ascending order.

This is a simple pen-and-paper game for two players which involves arcs and
nodes. It was invented in 1967 by Professor John H Conway and Michael S
Paterson and has been described and analysed in books and on the internet.
(Enter the words sprouts and Conway into a search engine for more information.)
Sprouts with three spots:
Draw three spots anywhere on the paper.
Each player in turn draws a line joining one spot to another spot (or itself) and
places a new spot somewhere on this line.
No lines may cross,
No spot may have more than three lines coming out of it (i.e. we would say that
the degree of any vertex cannot exceed three).
The game continues until no further moves are possible and the player to make
the final move is the winner.
A possible three spot game:
Analysis:
Providing students have met the fact that for any graph the node-sum is twice
the number of edges (from the hand-shake lemma), then it is quite easy to prove
that a three spot game of sprouts can never exceed 8 moves.
The game begins with 3 vertices and no edges.
Every move adds to the network 2 edges and one vertex.
Therefore after m moves, the network will have gained 2m edges and m vertices.
So the final graph contains 2m edges and m + 3 vertices.
Since the winning move will leave at least one vertex of degree 2, the maximum
number of 3-nodes is m + 2.
Therefore
node sum ≤ (m + 2) x 3 + 2
and since
node sum = 2 x (no. of edges)
we have
so
and so
2 x (2m) ≤ (m + 2) x 3 + 2
4m ≤ 3m + 8
m ≤ 8
Thus no game of three spot sprouts can exceed 8 moves.
Sprouts can be played with any number of spots and it can be shown that an n
spot game will never exceed 3n – 1 moves.
Brussels Sprouts:
A similar game can be played with crosses instead of spots, where each cross
represents a vertex of maximum degree 4. A move joins two ‘branches’ of
existing crosses (or a cross to itself) and a cross is placed on the new line.
The start of a possible two cross game:
In fact in Brussels Sprouts the first player always wins if the number of crosses
is odd and the second player always wins where the number of crosses is even!