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\begin{document}
\title{Early Writings on Graph Theory:\\Hamiltonian
Circuits and The Icosian Game}
\author{Janet Heine Barnett\footnote{Department of Mathematics and Physics;
Colorado State University - Pueblo; Pueblo, CO 81001 - 4901; {\tt janet.barnett@colostate-pueblo.edu}.}}
\date{}
\maketitle
\subsection*{Introduction}
Problems that are today considered to be part of modern graph theory originally appeared in a variety of
different connections and contexts. Some of these original questions appear little more than games or
puzzles. In the instance of the `Icosian Game', this observation seems quite literally true. Yet for the game's
inventor, the Icosian Game encapsulated deep mathematical ideas which we will explore in this project.
Sir William Rowan Hamilton (1805 - 1865) was a child prodigy with a
gift for both languages and mathematics. His academic talents were
fostered by his uncle James Hamilton, an Anglican clergyman with
whom he lived from the age of 3. Under his uncle's tutelage,
Hamilton mastered a large number of languages
--- including Latin, Greek, Hebrew, Persian, Arabic and Sanskrit
--- by the age of 10. His early interest in languages was soon eclipsed by his
interests in mathematics and physics, spurred in part by his contact
with an American calculating prodigy. Hamilton entered Trinity
College in Dublin in 1823, and quickly distinguished himself. He
was appointed Astronomer Royal of Ireland at the age of 22 based on
his early work in optics and dynamics. Highly regarded not only by
his nineteenth century colleagues, Hamilton is today recognized as a
leading mathematician and physicist of the nineteenth century.
In mathematics, Hamilton is best remembered for his creation of a new algebraic system known as the
`quarternions' in 1843. The system of quarternions consists of `numbers' of the form $Q = a + bi + cj + dk$
subject to certain basic `arithmetic' rules. The project that led Hamilton to the discovery of quarternions was
the search for an algebraic system that could be reasonably interpreted in the three-dimensional space of
physics, in a manner analogous to the interpretation of the algebra of complex numbers $a + bi$ in a
two-dimensional plane. Although this geometrical interpretation of the complex numbers is now standard, it was
discovered by mathematicians only in the early 1800's and thus was relatively new in Hamilton's time. Hamilton
was one of several nineteenth century British mathematicians interested in developing a purely \emph{algebraic}
foundation for complex numbers that would capture the essence of this geometrical interpretation. His algebraic
development of the complex numbers as ordered pairs of real numbers $(a,b)$ subject to certain operations
appeared in a landmark 1837 essay entitled \emph{Theory of Conjugate Functions, or Algebraic Couples; with a
Preliminary and Elementary Essay on Algebra as the Science of Pure Time}.
Hamilton concluded his 1837 essay with a statement concerning his
hope that he would soon publish a similar work on the algebra of
triplets. After years of unsuccessful work on this problem,
Hamilton was able to solve it in 1843 only by abandoning the
property of commutativity. For example, two of the basic
multiplication rules of the quarternion system are $ij =k$ and $ji =
-k$, so that $ij \neq ji$. Hamilton also replaced `triplets' by the
`four-dimensional' quarternion $a + bi + cj + dk$. Soon after
Hamilton's discovery, physicists realized that only the `vector
part' $bi+cj+dk$ of a quarternion was needed to represent
three-dimensional space. Although vectors replaced the use of
quarternions in physics by the end of the nineteenth century, the
algebraic system of vectors retains the non-commutativity of
quarternions.
Today's students of mathematics are familiar with a variety of non-commutative algebraic operations, including
vector cross-product and matrix multiplication. In Hamilton's day, however, quarternions constituted a major
breakthrough comparable to the discovery of non-Euclidean geometry. Immediately following Hamilton's 1843
announcement of his discovery, at least seven other new numbers systems were discovered by several other
British algebraists. In the `Icosian Game', Hamilton himself developed yet another example of a non-commutative
algebraic system. In this project, we explore both the algebra of that system and the graph theoretical notion
of `Hamiltonian circuit' on which Hamilton's interpretation of this algebra is based. The idea for the game was
first exhibited by Hamilton at an 1857 meeting of the British Association in Dublin \cite{JBHamilton}, and later
sold for 25 pounds to `John Jacques and Son,' a wholesale dealer in games. We begin with the preface to the
instructions pamphlet which Hamilton prepared for marketing of the game in 1859 \cite[pp. 32 - 35]{JBBiggs}.
\subsection*{The Icosian Game and Hamiltonian Circuits}
%Start Hamilton's original text.
\vspace{10pt}
\begin{quote}
\centerline{\textsf{\bf{THE ICOSIAN GAME}}}
\vspace{5pt}
\vspace{10pt}
\begin{figure}[htb]
\centerline{\epsfig{figure=icosian1.eps,width=3in}}
\end{figure}
\vspace{10pt}
\textsf{In this new Game (invented by Sir WILLIAM
ROWAN HAMILTON, LL.D., \&c., of Dublin, and by him named
\textit{Icosian} from a Greek word signifying `twenty') a player is
to place the whole or part of a set of twenty numbered pieces or men
upon the points or in the holes of a board, represented by the
diagram above drawn, in such a manner as always to proceed
\emph{along the lines} of the figure, and also to fulfil certain
\emph{other} conditions, which may in various ways be assigned by
another player. Ingenuity and skill may thus be exercised in
\emph{proposing} as well as in \emph{resolving} problems of the
game. For example, the first of the two players may place the first
five pieces in any five consecutive holes, and then require the
second player to place the remaining fifteen men consecutively in
such a manner that the succession may be \emph{cyclical}, that is,
so that No. 20 may be adjacent to No. 1; and it is always possible
to answer any question of this kind. Thus, if B C D F G be the five
given initial points, it is allowed to complete the succession by
following the alphabetical order of the twenty consonants, as
suggested by the diagram itself; but after placing the piece No. 6
in hole H, as before, it is \emph{also} allowed (by the supposed
conditions) to put No. 7 in X instead of J, and then to conclude
with the succession, W R S T V J K L M N P Q Z.
Other Examples of Icosian Problems, with solutions of some of them,
will be found in the following page.}
\end{quote}
\vspace{10pt}
In graph theoretic terminology, the holes of the game board are
referred to as \emph{vertices} (singular: \emph{vertex}) and the
lines that join two holes (vertices) are called \emph{edges}. The
collection of vertices and edges in a given relationship (as
represented by a diagram such as the game board) is called a
\emph{graph}. Two vertices that are joined by an edge in the graph
are said to be \emph{adjacent}. Thus, the instruction
`\textsf{always to proceed \emph{along the lines} of the figure'}
requires the player to find a sequence of adjacent vertices; such a
sequence is known as a \emph{path}. In the case where no vertex is
repeated in the sequence, the path is said to be a \emph{simple
path}. In the case where every vertex of the graph is used exactly
once in the sequence, the path is said to be a \emph{Hamiltonian
path}. The term \emph{cycle} is now used to describe what Hamilton
referred to as a `cyclical' path.
%Project Questions.
\bigskip
\begin{enumerate}
\item[1.] Explain why the rules of the
Icosian Game require players to always find
a simple path.
\item[2.] Use modern terminology to formally define
the terms \emph{cycle} and \emph{Hamiltonian cycle}.
\end{enumerate}
\vspace{10pt}
Following the preface, Hamilton includes several examples of Icosian
Problems in the Instruction Pamphlet. We consider only the first
two problems as a means of familiarizing ourselves with the concepts
of Hamiltonian cycle and Hamiltonian path.
\bigskip
%Continue with Hamilton's text.
\begin{quote}
\centerline{\textsf{\bf{EXAMPLES OF ICOSIAN PROBLEMS}}}
\vspace{5pt}
\centerline{\textsf{FIRST PROBLEM}}
\textsf{\emph{Five} initial points are given; cover the board, and
finish \emph{cyclically}. (As hinted in the preceding page, a
succession is said to be \emph{cyclical} when the \emph{last} piece
is adjacent to the \emph{first}.)}
\textsf{[This problem is always possible in at least two, and
sometimes in four, different ways. Two examples have been assigned:
the following are a few others.]}
\textsf{\emph{Example 3.} Given B C P N M as initial: two
solutions exist; one is the succession, D F K L T S R Q Z X W V J H
G; the other is D F G H X W V J K L T S R Q Z.}
\textsf{\emph{Example 4.} Five initials, L T S R Q. Four
solutions.}
\textsf{\emph{Example 5.} Five initials, J V T S R. Two
solutions.}
\end{quote}
\bigskip
%Project Questions.
\begin{enumerate}
\item[3.] Explain why Hamilton's first problem is equivalent to
the problem of finding a Hamilton circuit beginning from a given
initial sequence of five vertices.
\item[4.] In \textsf{\emph{Example 3}},
Hamilton specifies $ B \, C \, P \, N \, M $ as the first five
vertices in the desired circuit. He then claims that the two
solutions listed in the example are the \emph{only} two solutions of
this particular problem. Prove that these are in fact the only two
solutions by completing the details of the following argument.
Include copies of the diagram illustrating each step of the argument
in a different color as part of your proof.
\begin{enumerate}
\item Explain why the initial conditions for this example imply
that the solution to the problem must include either the sequence $
R \, S \, T$ or the sequence $ T \, S \, R$.
\item Explain why the initial conditions for this example imply
that the solution to the problem must include either the sequence $
R \, Q \, Z$ or the sequence $ Z \, Q \, R$.
\item Explain why we can now conclude that the solution to this
problem must include either the sequence $ X \, W \, V$ or the
sequence $ V \, W\, X$.
\item Explain why the initial conditions for this example imply
that the solution to the problem must include either the sequence $
F \, D \, M$ or the sequence $ M \, D \, F$.
\item Explain why we can now conclude that the solution to this
problem must include either the sequence $ K \, L \, T$ or the
sequence $ T \, L\, K$.
\item Use the information from above concerning which edges and
vertices we know must be part of the solution to prove that the two
circuits Hamilton lists are the only to solutions to the problem.
\end{enumerate}
\item[5.] In \textsf{\emph{Example 4}},
Hamilton claims there are four Hamiltonian circuits that begin with
the vertices $ L \, T \, S \, R \, Q $. Find them. (You do not
need to prove these are the only four.)
\end{enumerate}
\vspace{10pt}
\noindent Although Hamilton claims that every initial sequence
of five vertices will lead to at least two solutions (and possibly
four) within the Icosian Game, he does not offer a proof of this
claim. Nor does he claim that this is true of all graphs.
\bigskip
\begin{enumerate}
\item [6.] Show that it is not true of every graph
that any initial sequence of five vertices will lead to at least one
Hamiltonian circuit by finding an example of a graph with at least 5
vertices that has no Hamiltonian circuit. Prove that your graph
does not contain a Hamiltonian circuit.
\end{enumerate}
\bigskip
\noindent The next question pertains to Hamilton's
second problem, which Hamilton describes in the pamphlet as follows:
\vspace{10pt}
%Continue with Hamilton's text.
\begin{quote}
\centerline{\textsf{\bf{EXAMPLES OF ICOSIAN PROBLEMS }(continued)}}
\vspace{5pt}
\centerline{\textsf{SECOND PROBLEM}}
\textsf{\emph{Three} initial points are given; cover the board
\emph{non-cyclically}. (A succession is said to be
\emph{non-cyclical} when the \emph{last} piece is \emph{not}
adjacent to the first.)}
\textsf{[This problem is sometimes soluble in only \emph{one} way;
sometimes in only \emph{two} ways; sometimes in \emph{four} ways;
and sometimes it is not soluble at all, as will be seen in the
following examples.]}
\textsf{\emph{Example 6.} Three initial points, B C D; cover, and
end with T. There is in this case only one solution, namely, F G H
X Z Q P N M L K J V W R S T. }
\textsf{\emph{Example 7.} Same initials; cover, and end with W.
Two solutions. }
\textsf{\emph{Example 8.} Same initials; cover, and end with J.
Two solutions. }
\textsf{[The same number of solutions exists, if it be required, having the same three initials, to end with K, or L, or N, or V.]}
\textsf{\emph{Example 9.} Same initials; cover, and end with R. Four
solutions. }
\textsf{\emph{Example 10.} Same initials; cover, and end with M.
Impossible. }
\textsf{[The same result, if it be required to end with F, or H, or Q, or S, or X.]}
\end{quote}
\bigskip
% Project Question
\begin{enumerate}
\item[7.] In \textsf{\emph{Example 10 }}, Hamilton claims the problem of
finding a `non-cyclical' path that uses all vertices beginning with
$ B \, C \, D $ and ending with $M$ is impossible. Prove that he
is correct.
\end{enumerate}
\bigskip
%Continue with Hamilton's text.
\subsection*{The Icosian Game and Non-Commutative Algebra}
We now turn to the portion of Hamilton's pamphlet which links the
Icosian Game to a non-commutative algebra.
\vspace{10pt}
\begin{quote}
\centerline{\textsf{\textbf{HINTS ON THE ICOSIAN CALCULUS,}}}
\centerline{\textsf{\textbf{OF WHICH THE ICOSIAN GAME IS DESIGNED TO
BE AN ILLUSTRATION.}}}
\textsf{I. In a "MEMORANDUM respecting a New System of Roots of
Unity,", which appeared in the \emph{Philosophical magazine} for
December 1856, Sir W. R. Hamilton expressed himself nearly as
follows (a few words only being here omitted):}
\textsf{`I have lately been led to the conception of a new system,
or rather \emph{family of systems,} of \emph{non-commutative roots
of unity}, which are entirely distinct from the \emph{i j k} of
quaternions, though having some general analogy thereto; and which
admit, even more easily than the quaternion symbols do, of
geometrical interpretation. In the system which seems at present to
be the most interesting one among those included in this new family,
I assume three symbols, $\iota , \kappa , \lambda$, such that
$\iota^2 = 1 , \kappa^3 = 1 , \lambda^5 = 1, \lambda = \iota\kappa$;
where $\iota\kappa$ must be \emph{distinguished} form $\kappa\iota$,
since otherwise we should have $\lambda^6=1 , \lambda=1$. As a very
simple \emph{specimen} of the symbolical conclusions deduced from
these fundamental assumptions I may mention that if we make
$\mu=\iota\kappa^2=\lambda\iota\lambda$, we shall have also $\mu^5 =
1$, $\lambda=\mu\iota\mu$; so that $\mu$ is a new fifth root of
reciprocity. A long train of such symbolical deductions is found to
follow; and every one of the results may be \emph{interpreted } as
having reference to the passage from \emph{face to face } (or from
corner to corner) of the \emph{icosahedron} (or of the
dodecahedron): on which account, I am at present disposed to give
the name of `Icosian Calculus' to this new system of symbols, and of
rules for their operations.'}
\end{quote}
\vspace{10pt}
The system of `\textsf{\emph{non-commutative roots of unity}}'
described above employs three symbols $\iota , \kappa , \lambda$
subject to the following (non-commutative) rules:
\[\iota^2 = 1 \, , \, \kappa^3 = 1 \, , \, \lambda^5 = 1 \, , \,
\lambda = \iota \kappa \, , \, \iota \kappa \neq \kappa \iota \]
\noindent The symbol `1' represents the identity, so that $1 \iota = \iota 1 = \iota $ , $1 \kappa = \kappa 1
= \kappa $ , and $1 \lambda = \lambda 1 = \lambda $. In Part II of {\textsf{Hints on the Icosian Calculus},
Hamilton describes in detail how to interpret his system of `non-commutative roots of unity' within the Icosian
Game. First, consider only the symbolic action of $\iota , \kappa , \lambda$ as defined by the above
multiplication rules to to complete Questions 8 and 9.
\bigskip
%Project questions.
\begin{enumerate}
\item[8.] Prove symbolically that $ \kappa
= \iota \lambda$.
\item[9.] Prove symbolically that $\iota \kappa ^2 = \lambda \iota
\lambda$.
(This shows that is makes sense to define the new symbol $\mu$ by
$\mu = \iota \kappa ^2 = \lambda \iota \lambda$.)
\end{enumerate}
\noindent Extra Credit Question: Show symbolically that $\mu ^ 5 = 1$.
\bigskip
We now consider Hamilton's interpretation of this algebraic system within the Icosian Game. This interpretation
provides a concrete method for deriving new symbolic equations such as those mentioned in the following excerpt.
\vspace{10pt}
%Continue with Hamilton's text.
\begin{quote}
\centerline{\textsf{\textbf{HINTS ON THE ICOSIAN CALCULUS}
(continued)}}
\textsf{II. In a LITHOGRAPH, which was distributed in Section A of the British Association, during its Meeting
at Dublin in 1857, Sir W. R. H. pointed out a few other symbolical results of the same kind: especially the
equations $\lambda\mu^2\lambda=\mu\lambda\mu$ , $\mu\lambda^2\mu=\lambda\mu\lambda$ ,
$\lambda\mu^3\lambda=\mu^2$ , $\mu\lambda^3\mu=\lambda^2$; and the formula $(\lambda^3\mu^3(\lambda\mu)^2)^2=1$,
which serves as a \emph{common mathematical type} for the solution of \emph{all cases} of the First Problem of
the Game. He also gave at the same time an oral (and hitherto unprinted) account of his rules of
\emph{interpretation} of the principal symbols; which rules, with reference to the present Icosian Diagram (or
ICOSIAN), may be briefly stated as follows:}
\begin{enumerate}
\item \textsf{The operation $\iota$ \emph{reverses} (or reads backwards) a
\emph{line} of the figure; changing, for example, B\,C to C\,B.}
\item \textsf{The operation $\kappa$ causes a line to \emph{turn} in a
particular direction round it final point; changing, for instance,
B\,C to D\,C.}
\item \textsf{The operation $\lambda$ changes a line considered as a \emph{side}
of a pentagon to the \emph{following side} thereof, proceeding
always \emph{right-handedly} for every pentagon except the large or
outer one; thus $\lambda$ changes B\,C to C\,D, but S\,R to R\,W.}
\item \textsf{The operation $\mu$ is \emph{contrasted} with $\lambda$, and
changes a line considered as a side of a \emph{different pentagon}, and in the \emph{opposite order} or
rotation, to the consecutive side of that \emph{other} pentagon; thus $\mu$ changes B\,C to C\,P, and S\,R to
R\,Q; but it changes also R\,S to S\,T, whereas $\lambda$ would change R\,S to S\,N.}
\item \textsf{The only operations employed in the \emph{game} are those
marked $\lambda$ and $\mu$; but another operation, $\omega=\lambda\mu\lambda\mu\lambda=\mu\lambda\mu\lambda\mu$,
having the property that $\omega^2=1$, was also mentioned in the Lithograph above referred to; and to complete
the present statement of \emph{interpretations}, it may be added that the effect of this operation $\omega$ is
to change an \emph{edge} of a pentagonal \emph{dodecahedron} to the \emph{opposite edge} of that \emph{solid};
for example, in the diagram, B\,C to T\,V.}
\end{enumerate}
\end{quote}
\bigskip
Note that proceeding \textsf{`right handedly'} may be described as moving clockwise around the appropriate
pentagon, so that action which is \textsf{`in the opposite order'} of proceeding \textsf{`right handedly'} may
be described as moving counter-clockwise.
%project Questions
\begin{enumerate}
\item[10.] Use the interpretation of $\iota$ in (1) to explain why
$\iota^2 = 1$.
Begin by looking at the effect of applying the operation $\iota$
\emph{twice in succession}, beginning with the edge $B\,C$. Then
explain in general.
\item[11.] Use the interpretation of $\kappa$ in (2) to explain why
$\kappa^3 = 1$.
Begin by looking at the effect of applying the
operation $\kappa$ \emph{three times in succession}, beginning with
the edge $B\,C$. Then look at the effect of applying the operation
$\kappa$ \emph{three times in succession}, beginning with the edge
$P\,N$. Finally, explain in general.
\item[12.] Use the interpretation of $\lambda$ in (3) to explain why
$\lambda^5 = 1$.
Begin by looking at the effect of applying the operation $\lambda$
\emph{five times in succession}, beginning with the edge $B\,C$.
Then look at the effect of applying the operation $\lambda$
\emph{five times in succession}, beginning with the edge $S\,R$.
Finally, explain in general.
\item[13.] Use the interpretation of $\mu$ in (4) to explain why $\mu^5
= 1$.
Begin by looking at the effect of applying the operation $\mu$
\emph{five times in succession}, beginning with the edge $B\,C$.
Then look at the effect of applying the operation $\mu$ \emph{five
times in succession}, beginning with the edge $R\, S$. Finally,
explain in general. (Note that this provides a geometric solution
of the extra credit question stated above, immediately following
question 9.)
\item[14.] Beginning with the edge $B\,C$, use the interpretations
given for the four symbols $\iota , \kappa , \lambda , \mu$ to
illustrate that $\mu = \iota \kappa ^2 = \lambda \iota \lambda$.
\end{enumerate}
\noindent Extra Credit Question: Establish the equation $\lambda\mu^2\lambda=\mu\lambda\mu$ symbolically; then
illustrate this equation within the Icosian Game, first beginning with the edge $B\,C$ and then beginning with
the edge $R\, S$.
\bigskip
\subsection*{Some Closing Remarks}
Notice Hamilton's claim that an \emph{algebraic proof} using equations of the type
$(\lambda^3\mu^3(\lambda\mu)^2)^2=1$ can be used to find all Hamiltonian cycles beginning with a specified
initial sequence of five vertices. Notice also the contrast between the graph theoretical proof that you
completed in question 4 above, and the proof that we would get of this same result by interpreting this degree
twenty equation within the context of the Icosian Game Board. In general, however, it is not viable to associate
an algebraic system with an arbitrary graph as a means to find all Hamiltonian circuits within that graph. In
fact, as you demonstrated in question 6 above, a graph may contain no Hamiltonian circuits at all. Unlike the
known situation for other kinds of circuits (e.g., Euler circuits), there is no known simple condition on a
graph which allows one to determine in all cases whether a Hamiltonian circuit exists or not. In the case that a
graph does contain a Hamiltonian circuit, we say the graph is \emph{Hamiltonian}.
The more general question of determining a condition under which a
graph is Hamiltonian was first studied by Thomas Penyngton Kirkman
(1840 - 1892). Unlike Hamilton, who was primarily interested in the
algebraic connections of one specific graph, Kirkman was
interested in the general study of `Hamiltonian
circuits' in arbitrary graphs. The rector of a small and isolated English parish,
Kirkman presented a paper on this subject to the Royal Society on 6
August 1855. Regrettably, his solution of the problem was
incorrect. He did, however, present a second paper in 1856 in which
he described a general class of graphs which do not contain such a
circuit. Kirkman also studied the existence of Hamiltonian circuits
on the dodecahedron, a variation of the Icosian Game which Hamilton also studied.
In fact, the two men met once in 1861 when Hamilton visited Kirkman at his
rectory. That Hamilton's name became associated with the
circuits, and not Kirkman's, appears to be one of the accidents of
history, or perhaps a credit to the fame of Hamilton's quarternions
and work in mathematical physics.
\section*{Notes to the Instructor}
This project contains two sub-sections ``The Icosian Game and Hamiltonian Circuits'' and ``The Icosian Game and
Non-Commutative Algebra,'' both of which were developed specifically for use in an introductory undergraduate
course in discrete mathematics. Because no prior background in graph theory is assumed, the connection to
symbolic algebra makes the project suitable for use in a junior-level abstract algebra course as well. In a
discrete mathematics course, the project could be assigned independently or in conjunction with one or both of
the projects ``Early Writings on Graph Theory: Euler Circuits and The K\"onigsberg Bridge Problem'' and ``Early
Writings on Graph Theory: Topological Connections,'' both of which appear in this volume. For students with no
prior knowledge of non-commutative algebras, the instructor may wish to provide more explicit directions for
Questions 8 and 9, or work these together as a whole class. Otherwise, the project may be completed by students
working in small groups over 2--3 in-class days, or assigned as a week-long individual project outside of class.
Multiple copies of the Icosian Game diagram will be needed for each student; use of color pencils or markers is
also highly recommended.
\begin{thebibliography}{99}
\bibitem{JBBiggs} Biggs, N., Lloyd, E., Wilson, R.,
\emph{Graph Theory: 1736--1936}, Clarendon Press, Oxford, 1976.
\bibitem{JBCrowe} Crowe, M. J., \emph{A History of Vector Analysis:
The Evolution of the Idea of a Vectorial System}, Dover Publications, New York, 1994.
\bibitem{JBHamilton} Hamilton, W. R., ``Account of the Icosian Game,''
{\em Proc. Roy. Irish. Acad.} {\bf 6} (1853-7), 415--416.
\bibitem{JBKatz} Katz, V., {\em A History of Mathematics: An
Introduction}, Second Edition, Addison-Wesley, New York, 1998.
\end{thebibliography}
\end{document}