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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.
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.
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