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Lesson 1: Combinations and Permutations
Step 3: The Rubik's Cube
Now invite each group to use a Rubik's cube as a manipulative and visual tool in solving some combinations and permutation problems.
Cube Basics
The Rubik's Cube three dimensions and six faces, each face has a different colour. The standard colours are red, orange, yellow, blue, green and white. Each face has nine facelets. The Rubik's cube is made up of smaller cubes, called cubies. There are corner cubies, edge cubies and centre cubies, which cannot be swapped with each other, ie. a corner cubie cannot become an edge piece. The Position occupied by a cubie is called a cubicle. Even in a scrambled state, the colour of a face is determined by the centre cubie, which does not rotate. There are twelve "moves" of the Rubik's cube: Clockwise turns of the front, back, left, right, top, bottom layers and their counterclockwise counterparts (or inverse moves).
The Rubik's Cube three dimensions and six faces, each face has a different colour. The standard colours are red, orange, yellow, blue, green and white. Each face has nine facelets. The Rubik's cube is made up of smaller cubes, called cubies. There are corner cubies, edge cubies and centre cubies, which cannot be swapped with each other, ie. a corner cubie cannot become an edge piece. The Position occupied by a cubie is called a cubicle. Even in a scrambled state, the colour of a face is determined by the centre cubie, which does not rotate. There are twelve "moves" of the Rubik's cube: Clockwise turns of the front, back, left, right, top, bottom layers and their counterclockwise counterparts (or inverse moves).
Cube Exploratory:
- How many corner cubies does your Rubik's Cube have?
- How many facelets does each corner cubie have? There are eight corner cubies (with three facelets)
- How many edge pieces does it have?
- How many facelets does each edge cubie have? Twelve edge cubies (with two facelets)
- How about center cubies? six centre cubies (with one facelet)
- How many cubies are there in total? 26
Commentary
The Handbook of Cubik Math by Alexander H Frey, Jr. and David Singmaster is a good resource for a beginner's method to restoring the cube. OR the Cubers website lists online sources of cube instructions. Beginners should choose methods that restore one layer of the cube first, rather than solving the corners first method, which is for more advanced "speed cubers". Working on the cube is very motivating for students and it will deepen their understanding of the properties and movements of the cube, making the cube a more meaningful manipulative.
The Handbook of Cubik Math by Alexander H Frey, Jr. and David Singmaster is a good resource for a beginner's method to restoring the cube. OR the Cubers website lists online sources of cube instructions. Beginners should choose methods that restore one layer of the cube first, rather than solving the corners first method, which is for more advanced "speed cubers". Working on the cube is very motivating for students and it will deepen their understanding of the properties and movements of the cube, making the cube a more meaningful manipulative.
Question 1: Choose one face of your Rubik's Cube. There are many possible ways it could be scrambled. Look at the top row. How many different ways could these three cubies be arranged from left to right.
Answer: 63 = 6 x 6 x 6 = 216 ways (because repetitions of colour are allowed – this is similar to the combination lock example).
Question 2: How many different combinations of edge cubies could be located in one layer of the cube?
Answer: 12C4 = 12! / (12 – 4)!4! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x1 / (8 x 7 x 6 x5 x4 3 x 2 x 1) (4 x 3 x 2 x 1) = 12 x 11 x 10 x 9 / 4 x 3 x 2 x 1 (after cancellation) = 11880 / 24 = 495
2b: Why are we calculating the possibilities as combinations, rather than permutations?
Answer: Because we are dealing with only one layer, the 4 pieces we are selecting from twelve can go in any of the four cubicles, so there is no set order. This is not the case if you are considering arranging all of the edge pieces.
Question 3: How many different combinations of corner cubies could be located in one layer of the cube?
Answer: 8C4 = 8 x 7 x 6 x 5 / 4 x 3 x 2 x 1 = 1680 / 24 = 70
Now some tricky questions...
Students might want to use a calculator with a factorial and nCr, nPr functions. See the Technology Connection box at the end of the lesson.
Question 4: How many different possible states are there for all 8 corner pieces?
Hints: Now we are dealing with arranging all the corners on the cube. Remember to use the multiplication rule to take into account the arrangement of all 8 corner pieces, and each of their three possible orientations. Also take into account the limitations of the cube, in that when you are solving all the corners, the orientation of the last piece is determined by how you positioned the other 7. You cannot rotate the one remaining corner.
Answer: 8! X 3^7 = (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) x (3 x 3 x 3 x 3 x 3 x 3 x 3) = 423,360 x 2187 = 88,179,840, because you're permuting 8 items, and each has 3 possible states (3 possible "rotations" or "flippages"). It’s only 3^7 because the orientation of the last corner piece is determined by the other 7. (i.e., once you place the other 7 corner pieces, you can't flip just one corner piece).
Question 5: How many different possible states are there for all 12 edge pieces?
Answer: 12! * 2^11 = (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) x (2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2), because you're permuting 12 items, and each has 2 possible states (2 possible "rotations" or "flippages"). It's only 2^11 because the orientation of the last edge piece is determined by the other 11. (i.e., you can't flip just one edge piece).
The Ultimate Question 6a: How many total possible states can the Rubik’s cube have? Use the multiplication rule to combine your edge piece calculations with your corner piece calculations, and remember that the centre cubies do not move, so they cannot be rearranged.
Answer: 8! X 3^7 x 12! X 2^11 / 2 = = (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) x (3 x 3 x 3 x 3 x 3 x 3 x 3) x (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) x (2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2) / 2. = 43,252,003,274,489,856,000 (that's 43 Quintillion)
Question 6b: Why do we divide anwser by two?
Answer: We divide by 2 for the final result, because every permutation of the cube has even parity (i.e. you can swap just two pieces, so 1/2 of the permutations are impossible).
