The Mathematics of the Rubik s Cube

Size: px

Start display at page:

Download "The Mathematics of the Rubik s Cube"

Ronald Wilson
6 years ago
Views:

1 In this lessn, students will explre the pssible number ways the pieces f a Rubik's Cube can be arranged, and still fit the criteria fr a Rubik's Cube. Clrs are riented in a set way, s sme pieces (such as a yellw-white edge) wuld nt exist n a Rubik's Cube. Students will find all pssible cmbinatins fr a 6-clred 3x3 cube, and then divide t take ut the 'impssible pieces.' Estimated lessn length: minutes Materials needed: One Rubik's Cube per student, r per small grup f students Student Handut (cpy ne fr each student) Calculatrs, ne per student r per small grup f students Lessn Outline: Once students have the apprpriate materials, they can read the student handut and fllw the directins.

2 I. Pssible Number f Cmbinatins f the Crner Cubes There are a ttal f crner cubes that can be in different lcatins arund the cube at any given time. We can chse a lcatin in 8! pssible ways. 8! Since the crner cube has different clrs, it can be rtated in 8 The ttal pssible cmbinatins f the crner cubes is 8! 3. This number equals. different variatins. S there are 3 8 different rientatins f a crner cubes altgether. II. Pssible Number f Cmbinatins f the Edge Cubes There are a ttal f edge cubes that can be in different lcatins arund the cube at any given time. We can chse a lcatin in! pssible ways. Since the edge cube has different clrs, it can be rtated in different variatins. S there are 12 different rientatins f the edge cubes altgether. The ttal pssible 12 cmbinatins f the edge cubes are!. This number equals. III. Ttal Number f Cmbinatins f the Entire Cube. T get ur final answer, we simply multiply the tw numbers we fund abve hwever, that wuld be ver-cunting. There is a small catch. It turns ut that yu can t turn the faces in such a way as t switch the psitins f tw cubes while returning all the thers t their riginal psitins. Therefre if yu get all but 2 cubes in place, there is nly ne attainable chice fr the psitins f the remaining 2 cubes. We must divide the prduct we fund abve by 2. Als If yu are slving the crner pieces, the very last crner piece can nly have 1 crrect rientatin s yu must divide the prduct abve by 3 t cancel ut the ther pssible variatins f that crner piece. And.. If yu are slving the edge pieces, the very last piece can have nly 1 crrect rientatin s yu must divide the prduct abve by 2 t cancel ut the ther pssible variatins f that edge piece. Okay we are ready fr ur final calculatin. Ttal Number f Pssibilities = ! 3 12! Write this number in scientific ntatin: The full number in standard frm is: This number in wrd frm wuld be:

3 By nw yu have figured ut that there are 43 quintillin 4.3x10 pssible cmbinatins f the cube. The questin that is ften asked is whether it is pssible fr smebdy t cme up with the slutin t the cube by making randm twists and turns. What is the prbability f that happening? When it cmes t slving the Rubik s cube by a randm series f twists and turns, yu have a better chance f. Bwling a perfect game (300) smetime in yur life (1 in 11,500) Becming a prfessinal athlete (1 in 22,000) Being struck by lightning tday. (1 in 576,000) Sptting a U.F.O. tday (1 in 3,000,000) Getting hit in the head frm falling parts ff f an airplane in yur lifetime (1 in 10,000,000) A meter landing n yur huse tday (1 in 182,138,880,000,000) Nw, if yu wanted t see all 4.3x10 different pssible cmbinatins f the cube hw lng wuld it take yu t d the twists? Let s assume that yu culd perfrm three twists per secnd. T determine the ttal number f secnds, yu wuld have t divide secnds 4.3x10 by 3. T determine the number f minutes, divide that number by 60. minutes T determine the number f hurs, divide that number by 60. hurs T determine the number f days, divide that number by 24. days T determine the number f years, divide that number by 365 years Interesting Fact: Accrding t scientists, the universe is x10 years ld.

4 ANSWER KEY I. Pssible Number f Cmbinatins f the Crner Cubes There are a ttal f 8 crner cubes that can be in 8 different lcatins arund the cube at any given time. We can chse a lcatin in 8! pssible ways. 8! Since the crner cube has 3 different clrs, it can be rtated in 8 different variatins. S there are 8 3 different rientatins f a crner cubes altgether. 8 The ttal pssible cmbinatins f the crner cubes is 8! 3. This number equals 264,539,520. II. Pssible Number f Cmbinatins f the Edge Cubes There are a ttal f 12 edge cubes that can be in 12 different lcatins arund the cube at any given time. We can chse a lcatin in 12_! pssible ways. Since the edge cube has 2 different clrs, it can be rtated in 12_ different variatins. S there are _2¹²_ different rientatins f the edge cubes altgether. The ttal pssible cmbinatins f the edge cubes are (_12!) ( 2¹²). This number equals _1,961,990,553,600_. III. Ttal Number f Cmbinatins f the Entire Cube. T get ur final answer, we simply multiply the tw numbers we fund abve hwever, that wuld be ver-cunting. There is a small catch. It turns ut that yu can t turn the faces in such a way as t switch the psitins f tw cubes while returning all the thers t their riginal psitins. Therefre if yu get all but 2 cubes in place, there is nly ne attainable chice fr the psitins f the remaining 2 cubes. We must divide the prduct we fund abve by 2. Als If yu are slving the crner pieces, the very last crner piece can nly have 1 crrect rientatin s yu must divide the prduct abve by 3 t cancel ut the ther pssible variatins f that crner piece. And.. If yu are slving the edge pieces, the very last piece can have nly 1 crrect rientatin s yu must divide the prduct abve by 2 t cancel ut the ther pssible variatins f that edge piece. Okay we are ready fr ur final calculatin. Ttal Number f Pssibilities = Write this number in scientific ntatin: _4.3 x ! 3 12! The full number is: 43,252,003,274,489,596,000 This number in wrd frm wuld be frty-three quintillin, tw hundred fifty-tw quadrillin, three trillin, tw hundred seventy-fur billin, fur hundred eighty-nine millin, five hundred ninety-six thusand.

5 By nw yu have figured ut that there are 43 quintillin 4.3x10 pssible cmbinatins f the cube. The questin that is ften asked is whether it is pssible fr smebdy t cme up with the slutin t the cube by making randm twists and turns. What is the prbability f that happening? When it cmes t slving the Rubik s cube by a randm series f twists and turns, yu have a better chance f. Bwling a perfect game (300) smetime in yur life (1 in 11,500) Becming a prfessinal athlete (1 in 22,000) Being struck by lightning tday (1 in 576,000) Sptting a U.F.O. tday (1 in 3,000,000) Getting hit in the head frm falling parts ff f an airplane in yur lifetime (1 in 10,000,000) A meter landing n yur huse tday (1 in 182,138,880,000,000) Nw, if yu wanted t see all 4.3x10 different pssible cmbinatins f the cube hw lng wuld it take yu t d the twists? Let s assume that yu culd perfrm three twists per secnd. T determine the ttal number f secnds, yu wuld have t divide 1.44 X 10 secnds 4.3x10 by 3. T determine the number f minutes, divide that number by X minutes T determine the number f hurs, divide that number by X hurs T determine the number f days, divide that number by X days T determine the number f years, divide that number by X years

6 Natinal Standards Instructinal prgrams frm prekindergarten thrugh grade 12 shuld enable all students t: Understand numbers, ways f representing numbers, relatinships amng numbers, and number systems **Grades 6 8 Expectatins: In grades 6 8 all students shuld develp an understanding f large numbers and recgnize and apprpriately use expnential, scientific, and calculatr ntatin; use factrs, multiples, prime factrizatin, and relatively prime numbers t slve prblems; **Nte t Teacher Regarding Cmmn Cre and Natinal Standard Alignment Althugh this lessn is aligned with 7th and 8th grade level Cmmn Cre Standards, and 6-8 grade Natinal Math Standard Expectatins, yu will find the cncepts taught in this lessn such as factrials, scientific ntatin, and prbability are fund within sme High Schl level mathematics curriculum. Cmmn Cre Standards** GRADE 8 8.EE Expressins and Equatins Wrk with radicals and integer expnents. 2. Use square rt and cube rt symbls t represent slutins t equatins f the frm x²=p and x³=p, where p is a psitive ratinal number. Evaluate square rts f small prefect squares and cube rts f small perfect cubes. Knw that 2 is irratinal. 3. Use numbers expressed in the frm f a single digit times an integer pwer f 10 t estimate very large r very small quantities, and t express hw many times as much ne is than the ther. 4. Perfrm peratins with numbers expressed in scientific ntatin, including prblems where bth decimal and scientific ntatin are used. Use scientific ntatin and chse units f apprpriate size fr measurements f very large r very small quantities (e.g., use millimeters per year fr seaflr spreading). Interpret scientific ntatin that has been generated by technlgy.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Yu will learn the fllwing in this lab: The UNIVERSITY f NORTH CAROLINA at CHAPEL HILL Cmp 541 Digital Lgic and Cmputer Design Prf. Mntek Singh Fall 2016 Lab Prject (PART A): Attaching a Display t the Prcessr