The Mathematics of the Rubik’s Cube
Lesson submitted by Jim Pardun, Math Teacher, Vernon Hills HS. EdVenture Group added standards and answer sheet.
National Standards
Instructional programs from prekindergarten through grade 12 should
enable all students to:
Understand numbers, ways of representing numbers, relationships
among numbers, and number systems
**Grades 6–8 Expectations: In grades 6–8 all students should–
o
o
develop an understanding of large numbers and recognize and
appropriately use exponential, scientific, and calculator notation;
use factors, multiples, prime factorization, and relatively prime
numbers to solve problems;
**Note to Teacher Regarding Common Core and National Standard Alignment
Although this lesson is aligned with 7th and 8th grade level Common Core Standards, and 6-8 grade National
Math Standard Expectations, you will find the concepts taught in this lesson such as factorials, scientific
notation, and probability are found within some High School level mathematics curriculum.
Common Core Standards**
GRADE 8
8.EE Expressions and Equations
Work with radicals and integer exponents.
2. Use square root and cube root symbols to represent solutions to equations of the form x²=p and
x³=p, where p is a positive rational number. Evaluate square roots of small prefect squares and
cube roots of small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very
large or very small quantities, and to express how many times as much one is than the other.
4. Perform operations with numbers expressed in scientific notation, including problems where both
decimal and scientific notation are used. Use scientific notation and choose units of appropriate size
for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor
spreading). Interpret scientific notation that has been generated by technology.
GRADE 7
7.SP Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models.
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the
fraction of outcomes in the sample space for which the compound event occurs.
I. Possible Number of Combinations of the Corner Cubes
•
There are a total of _______ corner cubes that can be in ______ different locations around the
cube at any given time. We can choose a location in 8! possible ways.
(8! = 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1) Since the corner cube has ______ different colors, it can be rotated in
______ different variations. So there are 38 different orientations of a corner cubes altogether.
© 2011 Seven Towns Limited
Rubik’s ® is a registered trademark of Seven Towns, Ltd. London, England
www.youcandothecube.com
The Mathematics of the Rubik’s Cube
Lesson submitted by Jim Pardun, Math Teacher, Vernon Hills HS. EdVenture Group added standards and answer sheet.
( )
The total possible combinations of the corner cubes is ( 8!) 38 . This number equals
__________________.
II. Possible Number of Combinations of the Edge Cubes
•
There are a total of _________ edge cubes that can be in _______ different locations around
the cube at any given time. We can choose a location in _____! possible ways. Since the
edge cube has _______ different colors, it can be rotated in _____ different variations. So
12
there are ____ different orientations of the edge cubes altogether. The total possible
(
)
combinations of the edge cubes are ( ____!) ____12 . This number equals
_________________.
III. Total Number of Combinations of the Entire Cube.
•
To get our final answer, we simply multiply the two numbers we found above however, that
would be over-counting. There is a small catch.
o It turns out that you can’t turn the faces in such a way as to switch the positions of two
cubes while returning all the others to their original positions. Therefore if you get all
but 2 cubes in place, there is only one attainable choice for the positions of the
remaining 2 cubes. We must divide the product we found above by 2. Also…
o If you are solving the corner pieces, the very last corner piece can only have 1 correct
orientation so you must divide the product above by 3 to cancel out the other possible
variations of that corner piece. And…..
o If you are solving the edge pieces, the very last piece can have only 1 correct
orientation so you must divide the product above by 2 to cancel out the other possible
variations of that edge piece.
•
Okay… we are ready for our final calculation.
Total Number of Possibilities =
(8!) ( 38 ) (12!) ( 212 )
( 2 )( 3)( 2)
Write this number in scientific notation: _________________________
This number said verbally would be 43 quintillion combinations.
© 2011 Seven Towns Limited
Rubik’s ® is a registered trademark of Seven Towns, Ltd. London, England
www.youcandothecube.com
The Mathematics of the Rubik’s Cube
Lesson submitted by Jim Pardun, Math Teacher, Vernon Hills HS. EdVenture Group added standards and answer sheet.
(
)
By now you have figured out that there are 43 quintillion 4.3x1019 possible combinations of the
cube. The question that is often asked is whether it is possible for somebody to come up with
the solution to the cube by making random twists and turns. What is the probability of that
happening?
When it comes to solving the Rubik’s cube by a random series of twists and turns, you have a
better chance of….
o
Bowling a perfect game (300) sometime in your life (1 in 11,500)
o
Becoming a professional athlete (1 in 22,000)
o
Being struck by lightning today. (1 in 576,000)
o
Spotting a U.F.O. today (1 in 3,000,000)
o
Getting hit in the head from falling parts off of an airplane in your lifetime (1 in 10,000,000)
o
A meteor landing on your house today (1 in 182,138,880,000,000)
Now, if you wanted to see all 4.3x1019 different possible combinations of the cube how long would
it take you to do the twists? Let’s assume that you could perform three twists per second.
o
To determine the total number of seconds, you would have to divide 4.3x1019 by 3.
____________seconds
o
To determine the number of minutes, divide that number by 60.
____________ minutes
o
To determine the number of hours, divide that number by 60.
____________ hours
o
To determine the number of days, divide that number by 24.
____________days
o
To determine the number of years, divide that number by 365
____________ years
Interesting Fact:
According to scientists, the universe is
1.37x1010 years old.
© 2011 Seven Towns Limited
Rubik’s ® is a registered trademark of Seven Towns, Ltd. London, England
www.youcandothecube.com
The Mathematics of the Rubik’s Cube
Lesson submitted by Jim Pardun, Math Teacher, Vernon Hills HS. EdVenture Group added standards and answer sheet.
ANSWER KEY
I. Possible Number of Combinations of the Corner Cubes
•
There are a total of ___8___ corner cubes that can be in __8__ different locations around the
cube at any given time. We can choose a location in 8! possible ways.
(8! = 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1) Since the corner cube has __3___ different colors, it can be rotated in
8
___8__ different variations. So there are 3 different orientations of a corner cubes altogether.
( )
8
The total possible combinations of the corner cubes is ( 8!) 3 . This number equals
__264,539,520_____.
II. Possible Number of Combinations of the Edge Cubes
•
There are a total of ___12____ edge cubes that can be in __12___ different locations around
the cube at any given time. We can choose a location in __12_! possible ways. Since the
edge cube has __2__ different colors, it can be rotated in __12_ different variations. So there
are _2¹²_ different orientations of the edge cubes altogether. The total possible combinations
of the edge cubes are (_12!) (__2¹²). This number equals _1,961,990,553,600_.
III. Total Number of Combinations of the Entire Cube.
•
To get our final answer, we simply multiply the two numbers we found above however, that
would be over-counting. There is a small catch.
o It turns out that you can’t turn the faces in such a way as to switch the positions of two
cubes while returning all the others to their original positions. Therefore if you get all
but 2 cubes in place, there is only one attainable choice for the positions of the
remaining 2 cubes. We must divide the product we found above by 2. Also…
o If you are solving the corner pieces, the very last corner piece can only have 1 correct
orientation so you must divide the product above by 3 to cancel out the other possible
variations of that corner piece. And…..
o If you are solving the edge pieces, the very last piece can have only 1 correct
orientation so you must divide the product above by 2 to cancel out the other possible
variations of that edge piece.
•
Okay… we are ready for our final calculation.
Total Number of Possibilities =
(8!) ( 38 ) (12!) ( 212 )
( 2 )( 3)( 2)
Write this number in scientific notation: _4.325200327449 x 1019
This number said verbally would be 43 quintillion combinations.
© 2011 Seven Towns Limited
Rubik’s ® is a registered trademark of Seven Towns, Ltd. London, England
www.youcandothecube.com
The Mathematics of the Rubik’s Cube
Lesson submitted by Jim Pardun, Math Teacher, Vernon Hills HS. EdVenture Group added standards and answer sheet.
(
)
By now you have figured out that there are 43 quintillion 4.3x1019 possible combinations of the
cube. The question that is often asked is whether it is possible for somebody to come up with
the solution to the cube by making random twists and turns. What is the probability of that
happening?
When it comes to solving the Rubik’s cube by a random series of twists and turns, you have a
better chance of….
o
Bowling a perfect game (300) sometime in your life (1 in 11,500)
o
Becoming a professional athlete (1 in 22,000)
o
Being struck by lightning today. (1 in 576,000)
o
Spotting a U.F.O. today (1 in 3,000,000)
o
Getting hit in the head from falling parts off of an airplane in your lifetime (1 in 10,000,000)
o
A meteor landing on your house today (1 in 182,138,880,000,000)
Now, if you wanted to see all 4.3x1019 different possible combinations of the cube how long would
it take you to do the twists? Let’s assume that you could perform three twists per second.
o
To determine the total number of seconds, you would have to divide 4.3x1019 by 3.
__1.44 X 1019__seconds
o
To determine the number of minutes, divide that number by 60.
__2.40 X 1017__ minutes
o
To determine the number of hours, divide that number by 60.
__4.00 X 1015__ hours
o
To determine the number of days, divide that number by 24.
__1.67 X 1014__days
o
To determine the number of years, divide that number by 365
__4.57 X 1011__ years
© 2011 Seven Towns Limited
Rubik’s ® is a registered trademark of Seven Towns, Ltd. London, England
www.youcandothecube.com