Solving Rubik's Cubes
with F#
Stachu Korick  @stachudotnet  hello@stachu.net
Nov 17, 2018
Talk Inspiration
 SpeedSolving Rubik's Cubes
 And other patternrelated hobbies
 Writing F#
A bit about me
Stachu
site: stachu.net
.NET
email: hello@stachu.net
github: @stachudotnet
facebook: @stachudotnet
twitter: @stachudotnet
The Plan
 [10] (this) Intro
 [10] OnRamp Domain Knowledge
 [15] Handson: How to represent and "turn" a cube?
 [15] Handson: How to scramble a Rubik's Cube?
 [15] Show off Inefficient Solvers
 (the Rubik's Cube is a "graph"!)
 [30] Discuss Efficient Solver Patterns
 (sure, but it's also a "group"!)
 [10] Show off SAFE usage (Fable+Elmish+Saturn)
 Handson opportunities!
 [10] Outro/Questions/Buffer
 (probably related to the Rubik's Cube speedsolving world)
A bit about the Rubik's Cube
My Cubing Background
My Cubing Background
The Physical Structure, Pt. 1
The core structure of the Rubik's Cube does not move relative to each
The Physical Structure, Pt. 2
12 Edges surround the core and the centers.
Each of these 12 pieces has two orientations; "correct" and "flipped."
An edge cannot have oppositeside colors.
The Physical Structure, Pt. 3
8 "corner" pieces surround the core and edges.
Each of them has 3 orientations: "correct," "twisted clockwise" and "twisted counterclockwise."
These also cannot have two "opposite" colors.
Basic 6 Moves
 R(
ight )  L(
eft )  U(p)
 D(own)
 F(
ront )  B(ack)
12 Derived Moves
 R is clockwise 90 degrees
 R' is counterclockwise 90 degrees.
 R2 is the R face turned 180 degrees.
 ...
An Example Algorithm
Referring to
Certain Pieces
Creating a Simple Solver

Modeling the cube's state

Writing a scrambler

Transcribing the basic 18 'moves'

Getting *some* solver running
Challenge!
How can you store the state of a Rubik's Cube?
Representing the Cube
Facelet
Facelet Array
Representing the Cube
Cubie
Piece Arrays
Representing the Cube
CO+CP+EO+EP
CO + CP + EO + EP
CO: 0 <= x1 < 3^7
EO: 0 <= x2< 2^11
CP: 0 <= x3< 8!
EP: 0 <= x4< 12!
Representing the Cube
Representing the Cube
Turning the Cube
Turning the Cube
Turning the Cube
Turning the Cube
Turning the Cube
Turning the Cube
Turning the Cube
Challenge!
Scramble a Rubik's Cube
Scrambling the Cube
Scrambling the Cube
The Cube is a Graph
R2 U2
R2
U2
U2 R2
U2 R2 U2 R2 U2 R2
U2 R2 U2
U2 R2 U2 R2
U2 R2 U2 R2 U2
R2 U2 R2
R2 U2 R2 U2
R2 U2 R2 U2 R2
DepthFirst Tree Searching
DepthFirst Tree Searching
IDA Tree Searching
Why is this so slow?
 43 quintillion is a big number. (really big)
 18^20 is a big number. (even bigger)
 `Execute` takes a while to change cube state
Creating a Better Solver
Some Tradeoffs for Consideration
 HumanLearnable
 Turning Ergonomics
 Support partiallydefined cubes?
 Move Count
 Resource Requirements
 CPU, RAM, Disk Storage
 Speed
Program Human Algorithms?
The "CFOP" Method
Cross
F2L
(First 2 Layers)
OLL
(Orientation of the Last Layer)
PLL
(Permutation of the Last Layer)
Creating a Complex Solver
focus: run in a reasonable amount of time
The Kociemba Algorithm
Phase 1
Phase 2
Solved:
 Edge Orientation
 Corner Orientation
 U/D Edges in U/D Layer
Solve the rest, using only <U,D,R2,L2,F2,B2>
Additional "tricks"
to Implement
 representing each of these components into simple 16bit integers
(16bits is a lot less than an int array!)  move depthcaching caching
(how many moves deep various positions are away from a 'solved' state)  caching positiontransitions
 storing the transitions between the 16bit numbers rather than computing them on the fly
 taking advantage of symmetries
 using suboptimal phase 1 solutions
Storing Cube State Cheaply
Corner Orientation
Storing Cube State Cheaply
Corner Permutation
Move Tables
Symmetry Reductions
Pruning Tables
function Treesearch(position p, depth d) if d = 0 then if p is solved then Hooray! else if d > 0 then if prune1[p] <= d and prune2[p] <= d then for each available move m Treesearch(result of m applied to p, d1)
Pruning Tables
Using SubOptimal Phase 1 Solutions
Let's See It in Action!
Running it all with SAFE
Thanks! Questions?
@stachudotnet on twitter and
(hello@)stachu.net
Topic suggestions:
 What other puzzles are featured at competitions?
 Can you tell us a cubing story or two?
 How do people solve them blindfolded?
 Are you ever going to record another WTF# episode?
Solving Rubik's Cubes with F#
By Stachu Korick
Solving Rubik's Cubes with F#
 884