God's Algorithm

God's Algorithm is the optimal solution from a puzzle state to another state, commonly the solved state. The term is sometimes used to refer to the algorithm itself, or an algorithmic procedure that finds such a solution efficiently.

God's Number

God's Number is either used to refer to the diameter of the group of the puzzle (the furthest distance two states can be from each other) or to the furthest distance any position can be from solved. God's Number has long been known for smaller puzzles, such as 2x2x2 and Pyraminx, but for 3x3x3 it was unknown until July 2010 and it is still unknown for bigger cubes (like the 4x4x4).

Table of God's Numbers

Puzzle	Metric	Lower Bound	Upper Bound	Conjectured Value	State Realizing Lower Bound	Source
2x2x2	HTM	11			R U' R2 U' F' R' U F2 R U' F	[1]
	QTM	14			U R U F' R' F R U F' R F U R' F	[2]
3x3x3	HTM	20			U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2	[3]
	QTM	26			U2 F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' D2	[4]
	STM	18	20	18		[5]
	ATM	16	20	16	R2 U' (F2 B') (U D') F' R' D B D' R' B U2 R' U2 F' R2	[6]
4x4x4	OBTM	35	55			[7], [8]
	SSTM	32	53			[9], [10]
	BTM	29	53			[11], [12]
5x5x5	OBTM	52	130			[13], [14]
NxNxN	OBTM	(1/4 log(24!/4!^6) + o(1)) N^2/log(N)	(c + o(1)) N^2/log(N)	(1/4 log(24!/4!^6) + o(1)) N^2/log(N)		[15]
Clock		12			DDUU u d6 DUDU u5' d4 UDUD u2' d5 UUDD u d6 UDDU u d UUDU u2 DUUU u2	[16], [17]
Megaminx	HTM	48	194			[18], [19]
	QTM	70				[20]
Pyraminx		11 (15 with tips)			U R U R' U L' R' L' U' L B' u l r b	[21]
Skewb		11			U L U' R' U B' U L R' U B	[22]
Square-1	Twist Metric	13			/ (0,-3) / (0,-1) / (1,0) / (-4,-4) / (0,-5) / (0,-4) / (-5,2) / (-4,0) / (-5,-2) / (0,1) / (5,0) / (0,3) / (3,0)	[23]
	FTM	31				[24]

External links

• God's number in HTM is 20
• God's number in QTM is 26
• God's algorithm in Wikipedia