# X1+X3+X5+...

Submitted by mdlazreg on Wed, 01/10/2007 - 18:02.

Hi,

Finding the diameter of God's Algorithm for the 3x3x3 cube is equivalent to finding the length of the sequence:

X0

X1

X2

X3

X4

X5

:

:

Xn

where X0=1; Xn is all positions reachable from the Xn-1 positions.

I have noticed and proved that for any rubik like puzzle , the following identity is true:

X1+X3+X5+....+X2n+1=X0+X2+X4+X6+....+X2n

This can be verified for the 2x2x2 cube sequence for example:

1

6

27

120

534

2,256

8,969

33,058

114,149

360,508

930,588

1,350,852

782,536

90,280

276

Though I have been reading many books and articles about rubik's like puzzles, I never came across the above identity. Did anyone see the above identity somewhere? Thanks.

mdlazreg

Finding the diameter of God's Algorithm for the 3x3x3 cube is equivalent to finding the length of the sequence:

X0

X1

X2

X3

X4

X5

:

:

Xn

where X0=1; Xn is all positions reachable from the Xn-1 positions.

I have noticed and proved that for any rubik like puzzle , the following identity is true:

X1+X3+X5+....+X2n+1=X0+X2+X4+X6+....+X2n

This can be verified for the 2x2x2 cube sequence for example:

1

6

27

120

534

2,256

8,969

33,058

114,149

360,508

930,588

1,350,852

782,536

90,280

276

Though I have been reading many books and articles about rubik's like puzzles, I never came across the above identity. Did anyone see the above identity somewhere? Thanks.

mdlazreg