Easy / Lucky / Hard / Weird scrambles Thread

1. 7.81 D' F2 D' L2 U' L2 R2 D' R2 U' R2 F' U F2 D' L R' F' L' U2 L2

x z' U L F' L' D' F D'//Xcross
U' y R U' R' U y R' U' R//F2L2
U' L' U L//F2L3
U' y L' U L U F R U' R' F'//VHLS
U R U' R2 D' r U2 r' D R2 U R'//COLL but I knew it would skip
 
1. 7.87 F2 D2 F2 L2 D2 R2 B' U2 F' D2 F2 D L' F' R U' L D2 R2 D'

x z' L U' R U R//cross + preserve pair
y' L' U L //insert
U' L U' L' U' L U' L' U L U' L'//pair 2
U2 y L' U L// pair 3
U' R U R'//pair 4
U' R' U2 R' D' R U2 R' D R2 U'//COLL EPLL skip
 
15.84 L2 F2 D' F2 U L2 U2 L2 U R2 F' U2 F R2 U' F U2 R' B2 U2

2 move cross with pair built. Rest of F2L was easy as well, didn't get a good time because I just woke up
 
Uhh, guys, I just got a square 1 scramble starting in a square shape. I am kinda new to square 1 so can any one tell me how rare this is?
Like, similar to a PLL skip or something?

Anyway, here's the scramble: (0, 5)/(0, 3)/(6, 3)/(-5, -5)/(0, -1)/(0, -3)/(-1, 0)/(-5, -4)/(0, -4)/(-4, 0)/(-2, 0)/(1, 0)/(3, 0)/(0, -5)

I got a PB 41.84, (Very slow) but I'm learning new algs at the moment so I'm not surprised

(edit) Beat my PB 2 solves later
 
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FastCubeMaster said:
Uhh, guys, I just got a square 1 scramble starting in a square shape. I am kinda new to square 1 so can any one tell me how rare this is?
Like, similar to a PLL skip or something?

Anyway, here's the scramble: (0, 5)/(0, 3)/(6, 3)/(-5, -5)/(0, -1)/(0, -3)/(-1, 0)/(-5, -4)/(0, -4)/(-4, 0)/(-2, 0)/(1, 0)/(3, 0)/(0, -5)

I got a PB 41.84, (Very slow) but I'm learning new algs at the moment so I'm not surprised

(edit) Beat my PB 2 solves later
Click to expand...

That's more like getting an LL skip, I've only ever gotten it once and I do SQ1 a lot
 
FastCubeMaster said:
Uhh, guys, I just got a square 1 scramble starting in a square shape. I am kinda new to square 1 so can any one tell me how rare this is?
Like, similar to a PLL skip or something?
Click to expand...

Hssandwich said:
I wouldn't say that it's like an LL skip, it's definitely more common than that, but it doesn't really have a 3x3 equivalent.
Click to expand...

YouCubing said:
That's more like getting an LL skip, I've only ever gotten it once and I do SQ1 a lot
Click to expand...

So, this is actually a harder thing to calculate than may seem on the surface, if we're using WCA-official scrambles only - the filter is pretty aggressive towards easy scrambles, which of course is weighted more against cubeshape skip than pretty much any other cubeshape. But if we allow _any_ scramble to be okay (which is what most random-state scramblers do), then we can do some math to find the technical probability out :)

Putting the math in a spoiler tag below, but the probability of getting a cubeshape skip is roughly 0.1% - 4/3678. For comparison, this is roughly 4.25 times rarer than an unforced OLL skip (1/216), ~17 times more common than an unforced LL skip (1/15552), and about 1.7 times more common than getting a <=2 move cross with no color neutrality (174/190080, according to this).

I would most compare it to getting a very easy cross/first block/eoline/etc., because cubeshape is generally the first step, and having that skip allows you to begin your solve on the next step.

So, since we're only caring about cubeshape, the puzzle we care about is nothing more than 8 corner pieces and 8 edge pieces. The only restriction to a valid state is that we can perform a / on it, which means that both the top and bottom can be cut in half. So, that means that, for each of the four halves, we can either have:

  • 3 corners, 0 edges - only one way to arrange these
  • 2 corners, 2 edges - there are (4 choose 2) = 6 ways to arrange the pieces here
  • 1 corner, 4 edges - there are (5 choose 1) = 5 ways to arrange the pieces here
  • 0 corners, 6 edges - only one way to arrange these


So, to find out the total number of legal cubeshapes (counting all mirrors, inverses, and ABFs), all we need to do is figure out how many ways we can put these half possibilities together into a legal state. We have 4 halves to choose from, and we know there must be exactly 8 corners and 8 edges; edges are more restrictive, so lets work with this:

  • (6,2,0,0) - One 6-edge half, one 2-edge half, 2 no-edge halves (example: do / 3,3 / 1,2 / 4,2 / -3,0 to set up). There are 4*3=12 ways to order these halves around (eg, is the 6-edge chunk on UL, UR, DL, DR, etc.), and the total number of possibilities for internal arrangements is 1*6*1*1 = 6. So there are 72 cases here.
  • (4,4,0,0) - two 4-edge halves, two 0-edge halves (example: / 3,3 / 1,2 / 4,2 /). There are (4 choose 2) = 6 ways to order the halves, and there are 5*5*1*1 = 25 internal arrangements. So, there are 150 cases here.
  • (4,2,2,0) - one 4-edge half, two 2-edge halves, one 0-edge half (ex: / 3,3 / 1,0 / 4,0). 4*3=12 ways for halves, 5*6*6*1=180 ways for internals, so 2160 total cases.
  • (2,2,2,2) - four 2-edge halves (ex: solved cubeshape). 1 way to arrange halves, 6^4 = 1296 ways for internals, so 1296 cases.

Adding these up, we get 72+150+2160+1296 = 3678 total cases. The total number of solved cubeshape cases we could get is 4 (the alignments (0,0), (1,0), (0,-1), and (1,-1)), so the probability of getting a cubeshape skip is therefore 4/3678 = 0.0010875... ~ 0.1%.

These numbers are confirmed by looking at the TNoodle (WCA-official) code for Square-1, and seeing that Chen Shuang also uses 13 to represent half-layer information (1+5+6+1 = 13), and 3678 to represent the total number of possible cubeshapes.
 
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