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\bibitem[Ar]{Ar} M. Artin, {\bf Algebra}, Prentice-Hall, 1991

\bibitem[AM]{AM} E. Assmus, Jr. and H. Mattson, ``On the
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{\bf Combinatorial mathematics and its applications}, ed. R. Bose, 
T. Dowling, Univ of North Carolina Press, Chapel Hill,
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\bibitem[A]{A} 
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, 
R. A. Wilson, {\bf Atlas of Finite Groups}, 
Clarendon Press, Oxford, 1985.

\bibitem[Ba]{Ba} J. Baez, ``Some thoughts on the number 6'', 
internet newsgroup sci.math article, posted May 22, 1992, 
\newline
\verb+http://math.ucr.edu/home/baez/six.html+

\bibitem[B]{B} C. Bandelow, 
{\bf Inside Rubik's cube and beyond}, Birkh\"auser, 
Boston, 1980 

\bibitem[BH]{BH} R. Banerji and D. Hecker, 
``The slice group in Rubik's cube'',
Math. Mag. \underbar{58}(1985)211-218


\bibitem[BCG]{BCG} E. Berlekamp, J. Conway, R. Guy, 
%{\bf Winning ways for yorr mathematical plays, vol. 4},
%AK Peters, 2Rev Ed edition, 2004.
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\bibitem[BFR]{BFR} E. Bonsdorff, K. Fabel, O. Riihimaa, 
{\bf Schach und zahl}, Walter Rau Verlag, D\"usseldorf, 1966

\bibitem[BLS]{BLS}
A. Bj\"orner, L. Lovasz, and P. Shor,
``Chip-firing games on graphs,'' European J. of Combin.
\underline{12}(1991)283-291

\bibitem[Bu]{Bu} G. Butler, {\bf Fundamental algorithms for
permutation groups}, Springer-Verlag, Lecture Notes in Computer Science, 
\underbar{559}, 1991.

\bibitem[Ca]{Ca} R. Calinger, {\bf Classics of mathematics},
Prentice Hall, NJ, 1982.

\bibitem[Cam]{Cam}
L. Campbell, ``Orbix hints page,''
\newline
\verb+http://web.usna.navy.mil/~wdj/orbix_hints2.htm+

\bibitem[CD]{CD} M. Conrady and M. Dunivan,
``The Cross Group of the Rubik's Cube,'' USNA SM485C class project,
April,1997.
\newline
\verb+http://cadigweb.ew.usna.edu/~wdj/rubik/crossgp.html+
              
\bibitem[CS]{CS} J. Conway and N. Sloane, 
{\bf Sphere packings, lattices, and groups}, Springer-Verlag, 1993.

\bibitem[CS2]{CS2} 
------, 
``Lexicographic codes: error-correcting codes from game theory'', 
IEEE Transactions on Information Theory, \underbar{32}(1986)337-348.

\bibitem[CSW]{CSW} 
------ and A. Wilks, 
``Gray codes and reflection groups'', Graphs and combinatorics
\underbar{5}(1989)315-325 

\bibitem[CFS]{CFS} G. Cooperman, L. Finkelstein and
N. Sarawagi, ``Applications of Cayley graphs'', in 
{\bf Applied algebra ...}, Springer-Verlag, Lecture Notes in Computer 
Science, \underbar{508}, 1990

\bibitem[CK]{CK} D. Kunkle, G. Cooperman, ``Twenty-Six Moves 
Suffice for Rubik's Cube.'' Proceedings of the 
International Symposium on Symbolic and Algebraic
Computation (ISSAC '07), ACM Press. 

\bibitem[Cox]{Cox} H. S. M. Coxeter,
{\bf Regular polytopes}, Dover, 1973.

\bibitem[C]{C} J. Crossley, et al, 
{\bf What is mathematical logic?}, Dover, 1972

\bibitem[CL]{CL} Internet archives of the ``cube-lovers'' list at
\newline
\verb+http://www.permutationpuzzles.org/rubik/cube-lovers/+

\bibitem[CG]{CG} S. Curran and J. Gallian, ``Hamiltonian cycles and 
paths in Cayley graphs and diagraphs - survey,'' Discrete Math. 
\underline{156}(1996)1-18

\bibitem[DM]{DM} J. Davies and A. O. Morris, ``The
schur multiplier of the generalized symmetric group,''
J. London Math. Soc. \underline{8}(1974)615-620

\bibitem[DiMo]{DiMo} J. Dixon, B. Mortimer,
{\bf Permutation Groups},
Springer-Verlag,
Graduate Texts in Mathematics, 1996

\bibitem[El]{El} N. Elkies, ``On numbers and endgames: 
Combinatorial game theory in chess endgames,'' pages 135-150
in {\bf Games of no chance}, 
151--192, Math. Sci. Res. Inst. Publ., 29, 
Cambridge Univ. Press, Cambridge, 1996.)

\bibitem[E]{E2}
K. Eriksson, ``The numbers game and Coxeter groups,'' Discrete Math. 
\underline{139}(1995)155-166

\bibitem[EK]{EK} J. Ewing and C. Kosniowski, 
{\bf Puzzle it out, cubes, groups, and puzzles}, 
Cambridge Univ Press, 1982
 

\bibitem[FS]{FS} A. Frey and D. Singmaster, {\bf Handbook of cubik math}, 
Enslow Pub., 1982


\bibitem[G]{G} A. Gaglione, {\bf An introduction to group theory}, 
NRL, 1992
\newline
\verb+http:\\web.usna.navy.mil\~wdj\tonybook\index.html+

\bibitem[Gap]{Gap} 
The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.4.9;
2006. (\verb+http://www.gap-system.org+)

\bibitem[Gar1]{Gar1} M. Gardner, ``Combinatorial card problems''
in {\bf Time travel and other mathematical bewilderments},
W. H. Freeman, New York, 1988

\bibitem[Gar2]{Gar3} ------, 
``The binary Gray code'', in {\bf Knotted donuts and other
mathematical entertainments}, F. H. Freeman and
Co., NY, 1986 

\bibitem[GJ]{GJ} M. Garey and D. Johnson, {\bf
Computers and intractibility}, W. H. Freeman, New York, 1979

\bibitem[GT]{GT} K. Gold, E. Turner, ``Rubik's group'', 
Amer. Math. Monthly, \underbar{92}(1985)617-629


\bibitem[GKT]{GKT} J. Goldwasser and W. Klostermeyer
and G. Trapp, 
``Characterizing switch-setting problems'', 
Congressus Numerantium, vol. 126, 1997, pp. 99-111
(preprint available at
\newline
\verb+http://www.unf.edu/~wkloster/fib.html+.
)

\bibitem[GKTZ]{GKTZ} ---- and ---- and ---- and C. Zhang,
``Setting switches in a grid, '' 1995 preprint available at
\newline
\verb+http://www.unf.edu/~wkloster/fib.html+


\bibitem[Gr]{Gr} R. Grimaldi, {\bf Discrete and combinatorial 
mathematics}, $4^{th}$ ed., Addison-Wesley-Longman, 1999 

\bibitem[Gu]{Gu} GUAVA, a GAP package for error-correcting codes,
\newline
\verb+http://sage.math.washington.edu/home/wdj/guava/+

\bibitem[Hi]{Hi} R. Hill, {\bf A first course in coding theory},
Oxford Univ. Press, 1986

\bibitem[Ho]{Ho}
Dan Hoey post to cube-lover's list:
\newline
\verb+http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/+
\newline
\verb+Dan_Hoey__The_real_size_of_cube_space.html+

\bibitem[H]{H} D. Hofstater, {\bf Metamagical themas}, Basics
Books, 1985 (Mostly a collection of Scientific American
columns he wrote; the articles referred to here were also 
published in Scientific American, March 1981, July 1982)

\bibitem[HP]{HP} W. C. Huffman and V. Pless, {\bf Fundamentals of
error-correcting codes}, Cambridge Univ. Press, 2003.

\bibitem[Hum]{Hum} J. Humphreys, {\bf Reflection groups and coxeter 
groups}, Cambridge Univ Press, 1990

\bibitem[Ip]{Ip} IPython, \verb+http://ipython.scipy.org/+

\bibitem[I]{I} J. Isbell, ``The Gordon game of a finite group,''
Amer. Math. Monthly \underline{99}(1992)567-569

\bibitem[J1]{J} D. Joyner, ``Rainbow Masterball page,''
internet www page 
\begin{verbatim}
http://www.permutationpuzzles.org/mball/rainbow.html
\end{verbatim}

\bibitem[J2]{Jwww} ------, ``Permutation puzzle page,'' 
internet www page 
\begin{verbatim}
http://www.permutationpuzzles.org/
\end{verbatim}

\bibitem[JN]{JN} ----- and G. Nakos,
{\bf Linear algebra and applications}, Brook-Cole, 1998.

\bibitem[JKT]{JKT} ------, R. Kreminski,
J. Turisco,
{\bf Applied Abstract Algebra}, lecture notes,
to be published, www page 
\begin{verbatim}
http://cadigweb.ew.usna.edu/~wdj/book/index.html
\end{verbatim}

\bibitem[JS]{JS} ------ and A. Southern, ``The Masterball
puzzle'', preprint (at \cite{J})


\bibitem[Koc]{Koc} Herbert Kociemba, Cube solver site,
\verb+http://kociemba.org/cube.htm+

\bibitem[KG]{KG} ------ and Mike Godfrey posting:
\newline
\verb+http://cubezzz.homelinux.org/drupal/?q=node/view/22+

\bibitem[Ko]{Ko} T. W. K\"orner's {\bf The pleasure of counting},
Cambridge Univ. Press, 1996.

\bibitem[K]{K} B. Kostant, ``The graph of the
truncated icosahedron and the last letter of Galois,''
Notices of the A.M.S. \underbar{42}(1995)959-968

\bibitem[Lo]{Lo} M. Longridge, Cubeman's cube notes, 
\verb+http://cubeman.org/cubeman.html+

\bibitem[Lu]{Lu} A. Luers, 
``The group structure of the pyraminx and the dodecahedral 
faces of $M_{12}$'', USNA Honors thesis, 1997
(Advisor W. D. Joyner)
\newline
\verb+http://cadigweb.ew.usna.edu/~wdj/m_12.htm+

\bibitem[MT]{MT}
The MacTutor History of Mathematics archive
(maintained by John J O'Connor and Edmund F Robertson
at the School of Mathematical and Computational Sciences,
University of St Andrews, Scotland),
\newline
\verb+http://www-groups.dcs.st-and.ac.uk/~history/index.html+

\bibitem[MN]{MN} Peter Maier and Werner Nickel,
``Attainable patterns in Alien Tiles'',
American Mathematical Monthly, Volume 114, Number 1, January 2007 , pp. 1-13.

\bibitem[MS]{MS} F. MacWilliams and N. Sloane,
{\bf The theory of error-correcting codes}, 
North-Holland, 1977

\bibitem[MKS]{MKS} W. Magnus, A. Karrus and
D. Solitar, {\bf Combinatorial group theory}, 
2nd ed, Dover, 1976

\bibitem[N]{N} NetworkX graph theory program,
\verb+https://networkx.lanl.gov/+,

\bibitem[NST]{NST} P. Neumann, G. Stoy and E. Thompson, 
{\bf Groups and geometry}, Oxford Univ. Press, 1994

\bibitem[P]{P} D. Pelletier, ``Merlin's magic square,''
Amer. Math. Monthly \underline{94}(1987)143-150.

\bibitem[Pe]{Pe} M. Petkovi\'c, {\bf Mathematics and chess},
Dover, 1997

\bibitem[Py]{Py} Python programming language tutorial
(by Guido van Rossum),
\newline
\verb+http://www.python.org/doc/current/tut/tut.html+

\bibitem[Ra]{Ra} Silviu Radu,
``A New Upper Bound on Rubik's Cube Group,''
\newline
\verb+http://arxiv.org/abs/math.CO/0512485+

\bibitem[Re]{Re}  Michael Reid's Rubik's cube page,
\verb+http://www.math.ucf.edu/~reid/Rubik/index.html+

\bibitem[Ro]{Ro} T. Rothman, 
``Genius and biographers: the fictionalization of \'Evariste 
Galois,'' Amer. Math. Monthly \underbar{89} (1982)84--106

\bibitem[R]{R} J. J. Rotman, {\bf An introduction to the theory of groups},
4th ed, Springer-Verlag, Grad Texts in Math \underbar{148}, 1995

\bibitem[Ru]{Ru} E. Rubik, et al, 
{\bf Rubik's cubic compendium}, Oxford Univ Press, 1987

\bibitem[Rus]{Rus} D. Rusin's web pages on Lights Out,
\newline
\verb+http://www.math.niu.edu/~rusin/papers/uses-math/games/other/lights+.

\bibitem[S]{S} The \sage group,
\SAGE Mathematical Software, Version 2.7, 
\verb+http://www.sagemath.org/+.

\bibitem[Sa]{Sa} Dorothy Sayer, {\bf The nine tailors},
Harcourt Brace and Co., 1962

\bibitem[Sch1]{Ja} Jaap Scherphuis' Lights Out puzzle page, on the 
WWW at the URL:
\newline
\verb+http://www.geocities.com/jaapsch/puzzles/lights.htm+

\bibitem[Sch2]{Ja2} Jaap Scherphuis' puzzle page, on the 
WWW at the URL:
\newline
\verb+http://www.geocities.com/jaapsch/puzzles/+

\bibitem[Sch]{Sch} Martin Sch\"onert,
Analyzing Rubik's Cube with GAP webpage
\newline
\verb+http://www.gap-system.org/Doc/Examples/rubik.html+

\bibitem[Se]{Se} J.-P. Serre, {\bf Linear representations of finite
groups}, Springer-Verlag, 1977

\bibitem[Ser]{Ser} ------, {\bf Trees}, Springer-Verlag, 1980

\bibitem[Si]{Si} D. Singmaster, {\bf Notes on Rubik's 
magic cube}, Enslow, 1981

\bibitem[Sn1]{Sn} R. Snyder, {\bf Get cubed}, booklet, 1990

\bibitem[Sn2]{Sn2} ------, {\bf Turn to square 1}, booklet,
1993

\bibitem[So]{So} Leonard H. Soicher, GAP package GRAPH:
\verb+http://www.gap-system.org/Packages/grape.html+

\bibitem[Sti]{Sti} Lewis Benjamin Stiller,
{\bf Exploiting symmetries on parallel architecture},
PhD Dissertation, CS Dept, John Hopkins University, 1995 
\newline
(see also ``Multilinear algebra and chess endgames'',
in {\bf Games of no chance (Berkeley, CA,
1994)}, 151--192, Math. Sci. Res. Inst. Publ., 29, 
Cambridge Univ. Press, Cambridge, 1996)

\bibitem[Sto]{Sto} D. Stock, ``Merlin's magic square revisited,''
Amer. Math. Monthly \underline{96}(1989)608-610

\bibitem[St]{St} R. Stoll, {\bf Set theory and logic}, Dover, 1963

\bibitem[Ta]{Ta} 
J. Tawney, ``Turning the Lights Out in three
dimensions,'' Rose-Hulman Institute of Technology
Undergraduate Mathematics Journal, Volume 1, 2000 

\bibitem[TW]{TW} A. D. Thomas and G. V. Wood, {\bf Group tables},
Shiva Publishing Ltd, Kent, UK, 1980

\bibitem[Th]{Th} J. G. Thompson, ``Rational functions
associated to presentations of finite groups,'' 
J. of Algebra \underline{71}(1981)481-489

\bibitem[Tho]{Tho} T. Thompson,
{\bf From Error-Correcting Codes through Sphere Packings to Simple Groups},
Cambridge Univ. Press, 2004.

\bibitem[vLW]{vLW} J. van Lint and R. M. Wilson, {\bf A course in 
combinatorics},  Cambridge Univ. Press, 1992


\bibitem[We]{We} H. Weyl, 
{\bf Symmetry}, Princeton Univ Press, 1952

\bibitem[Wh]{Wh} White, Arthur, 
``Fabian Stedman: The First Group Theorist?'',
American Mathematical Monthly, Nov. 1996, pp771-778 

\bibitem[Wi]{Wi} Wikipedia,
\verb+http://wikipedia.org/+

\bibitem[Wba]{Wba} Wikipedia introduction to Boolean algebra:
\newline
\verb+http://en.wikipedia.org/wiki/Boolean_algebra+

\bibitem[Wbl]{Wbl} Wikipedia introduction to the ``Burrnside Lemma'':
\newline
\verb+http://en.wikipedia.org/wiki/Burnside's_lemma+


\bibitem[W]{W} R. M. Wilson, ``Graph puzzles, homotopy, and
the alternating group'', J. of Combin. Theory, \underbar{16}
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%\bibitem[]{} 
%\bibitem[]{} 
%\bibitem[]{} 



\end{thebibliography}