Friday, October 7, 2016

How to Solve the Rubik's Cube 2.0

When Art Benjamin visited our school, I bought his DVD called "The Mathematics of Puzzles: From Cards to Sudoku." "Mastering the Rubik's Cube" was one of the 12 lectures on the DVD. I watched it this summer and was surprised how easy his method is. I broke it down for my students in my Problem-Solving class, and here are the 8 steps. I do think it's easier (and clearer instruction-wise) than my former blog post on solving the Rubik's Cube. Also, here is my post on why I teach it in my class and a link to the mosaic contest in which we won 3rd place.

8 Steps
First Layer
1. The Daisy
·      The yellow face is on top
·      White petals will surround the daisy.

2. The Easy (bottom edges)
·      The yellow face is on top
·      Twist cube so the white petal finds its center

·      Rotate face 180˚
·      Repeat for the other three white petals
·      Flip Rubik’s cube upside down and look at white cross

3. The 123 (bottom layer)
·      With white daisy on the bottom, find a white corner on the top rim
·      Twist it until its other color finds it’s center (it matches in color with the center color)
·      Hold the cube so that this white cube is on a side.
·      ONE: Point with your index finger to the white. If you’re pointing with your right hand, you will twist the right face away from you. If you’re pointing with your left hand, you will twist the left face away from you. (UP)
·      TWO: Right hand, twist the top CLOCKWISE. Left hand, twist the top COUNTERCLOCKWISE. Either way, whites will match up.
·      THREE: Right hand, bring right face back down toward you. Left hand, bring left face back down toward you. Either way, now white corner is in proper place on the “daisy.”
Note: a 1-2-3 always starts with an “up”

Complications:
What if there are no white corners on the top rim, but there is one in top face?
1. Rotate white so it’s directly above a non-white (unsolved) on bottom face.
2. Perform 1-2-2-3 move
3. Now the white is on the rim, so you can repair with the 1-2-3 move.

What if you don’t have any whites on a tip rim or face? Do a 1-2-3 move, and it will move a white piece into the top rim or face, and then you can repeat.

Second Layer
4. The Middle Layer
·      With white daisy on the bottom (or yellow face on top—same thing), practice this move:

 UP TWIST DOWN TWIST UP TWIST TWIST DOWN 1 2 3 2 1 2 2 3 R U R’ U R U U R’

(NOTE: the above does not have anything to do with the middle layer. It is just for practice for later)

·      Only have to change 4 edge pieces.
·      Start by looking at the top layer and see if there are any edge pieces on the rim that do not use the color yellow.
·      Find one that does not use the color yellow and twist it so that it matches its center color.
·      The color of the top edge will either match the left or right face. Whichever side it matches, give that side a mental “slap.”
·      Twist the top in the direction of the slap.
·      With same slapping hand, perform the 1-2-3 move.
·      Lost a white, fix it by moving the color on the white cubie to match its center.
·      Perform the 1-2-3 with the side that contained that white face (white should be on the side, NOT facing you.)

If all four-edge pieces have yellow, we need to move a piece out of its wrong place.
·      Find the piece that needs to be moved and perform a 1-2-3 move (no slapping required.)
·      Fix the white piece by moving it to match its center and follow with a 1-2-3 (note: the white will be on the side)

Third Layer
5. FUR U’R’F’ move (top cross)
·      F = front, B = back, R = right, L = left, U = up, and D = down
·      When you see these letters, always move the face clockwise
·      If you see these letters with a ‘ by it, always move the face counterclockwise, for example R’ is moving the right face counterclockwise.
·      Up face is always yellow, down face is always white
·      L moves toward you, R moves away from you
·      Perform at “9:00”
·      If you don’t have “9:00,” do FUR U’R’F’. You may have to do this twice, but then you want “9:00
·

·      Do FUR U’R’F’ one more time to get the yellow cross.

Just a reminder: R is up, R’ is down, L is down, L’ is up

6. The Fun move (top face yellow)
·      Count the number of yellow corners on the top. Best case is one. You can only have zero, one, or two.
·      With 0 or 2 yellow corners, rotate the cube so that a yellow is in the upper left corner of the front face.

 UP TWIST DOWN TWIST UP TWIST TWIST DOWN 1 2 3 2 1 2 2 3 R U R’ U R U U R’
·      Repeat until you have only one yellow corner
·      Orient the cube so that the one yellow cube is in the bottom left corner
·      Repeat the “fun” move and the one yellow cube in the bottom right corner until you have the top face yellow.

7. The R’F move (top corners)
·      Look around the top rim and see if any of the top sides have matching corner colors
·      If not, then do: R’ F R’ B2      R F’ R’ B2       R2
·      When you have matching corner pieces, move them to the back face
·      Repeat R’ F R’ B2      R F’ R’ B2       R2
·      Orient so that all corners should now be matched
(remember, R’ is “down”)

(or, know that Up or Down refers only to the right side, and the mantra is:
D F D B2      U F’ D B2       U2
8. The “FFURL” move (top edges)
·      All should be in proper orientation except for three or four edges.
·      If you only have three edges out of position, then you have one perfect side, which you want to move to the back
·      FFU   R’L   FF   RL’  UFF (or, FFU, both sides down, FF, both sides up, UFF)

All on one page:
1. The daisy
2. Bottom edges
3. The 123 move (bottom layer) (123 always starts with an up)
4. Middle layer (slap 123, adjust white 123)
5. FURU’R’F’ (look for 9:00)
7. R’ F R’ B2  R F’ R’ B2       R2 (where R’ is down) or D F D B2      U F’ D B2       U2
8. FFU, both sides down, FF, both sides up, UFF

I heard from a former student who is a freshman in college and who told me today, "It's so much fun to know...I always use it as a fun fact during ice breakers!" Love. I hope you'll have as much fun, too! My fastest time is 2:45, but normally it's around 3 minutes. There are definitely faster ways, but this method is, to me, easy to teach...as long as you have willing students.

My plan is to have my class teach some of the 5th graders at our school how to solve the first and possibly the second layers, so that maybe they will want to learn the rest on their own. I've joined a national Rubik's Cube club, and we may host a competition...we will see! Just like the Rubik's Cube, one step at a time :)

Gary White said...

Is accounting considered a math major?
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