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Mathematical
Puzzles
Mac Stannard
Stacia Orr
Mathematical Puzzles
A glimpse into the disconcertion of brilliance
Instant Insanity
Puzzle
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Mac Stannard
Stacia Orr
Department of Mathematics
Boise State University
April 2015
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Intro to Instant Insanity
Mathematical
Puzzles
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
Introduction
Graph theory
Reaching an end
game
• Created by Franz Owen Armbruster in 1967
• Goal: stack the cubes so that each color is seen exactly
once on the front, back, left, and right
• 41,472 possible arrangements
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Graph Theory
Mathematical
Puzzles
Mac Stannard
Stacia Orr
• We can solve this using graph theory
• Begin by mapping out the colors shown on each cube’s
three opposite sides
Instant Insanity
Puzzle
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Solution
Mathematical
Puzzles
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
• Break it into two separate graphs:
1. front/back
2. left/right
• Three criteria must be met for these graphs:
1. each graph must contain one edge from each cube
2. no edges are in common
3. vertices cannot be greater than degree three
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Solution
Mathematical
Puzzles
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
• Stack the cubes accordingly
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Group Theory
Mathematical
Puzzles
The mathematical study of structure and symmetry
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
• Permutation groups
• Cayley’s graph
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Mathematical
Puzzles
Meet the cube
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
• Cube anatomy
◦ faces and facelets
◦ cubies and cubicles
Introduction
Graph theory
Reaching an end
game
1. center
2. corner
3. edge
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
• Movement
◦ F, B, L, R, U, D
◦ direction
8! × 37 ×
12!
2
× 211
God’s number
Mathematical
Puzzles
The winning strategy of omniscience
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
• Fewest possible moves required to solve a Rubik’s cube
• God’s algorithm
• Holds for any arrangement
• Correlation to Cayley’s graph
• Presently known to be equal to 20
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
Mathematical
Puzzles
The Devil’s algorithm
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
• A set of moves that will eventually restore a cube to its
solved state
• The Devil’s number is presently believed to be between
34,326,986,725,785,600
and
43,251,683,287,486,463,996
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
For reflection and procrastination
Mathematical
Puzzles
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
Introduction
Graph theory
Reaching an end
game
• More puzzles
◦
◦
◦
◦
15 Puzzle
Pyraminx
Rainbow Masterball
Superflip
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading
For further reading
Mathematical
Puzzles
Mac Stannard
Stacia Orr
Instant Insanity
Puzzle
David Joyner.
Adventures in Group Theory.
The John Hopkins University Press, 2008.
Alexander H. Frey, Jr. and David Singmaster.
Handbook of Cubik Math.
Enslow Publishers, Inc., 1982.
M. Conrady and M. Dunivan.
The Cross Group of the Rubik’s Cube.
U.S. Naval Academy, 1997.
http://www.permutationpuzzles.org/rubik/crossgp.html
Introduction
Graph theory
Reaching an end
game
Rubik’s Magical
Cube
Group theory
God’s number and the
Devil’s algorithm
For Reflection and
Further Reading