Construction method for an order-64 Nasik & bimagic cube
December 5, 2009, Mitsutoshi Nakamura
An order-64 Nasik and bimagic cube M(x,y,z), where 0 <= x,y,z <= 63, 
is constructed by the following formula:
M(x,y,z) = (2^17)*D17 + (2^16)*D16 + ... + 2*D1 + D0 + 1,
x = (2^5)*x5 + (2^4)*x4 + (2^3)*x3 + (2^2)*x2 + 2*x1 + x0,
y = (2^5)*y5 + (2^4)*y4 + (2^3)*y3 + (2^2)*y2 + 2*y1 + y0,
z = (2^5)*z5 + (2^4)*z4 + (2^3)*z3 + (2^2)*z2 + 2*z1 + z0,
  D17     1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0     x5  
  D16     1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1     x4  
  D15     1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0     x3  
  D14     1 0 1 1 0 0 0 1 0 0 1 1 0 0 1 1 1 0     x2  
  D13     1 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1     x1  
  D12     1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1     x0  
  D11     0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0     y5  
  D10   =   0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0     y4   (mod. 2).
  D9     0 0 1 1 0 0 1 1 1 0 0 1 0 1 1 0 1 0     y3  
  D8     0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1     y2  
  D7     0 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0     y1  
  D6     0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1     y0  
  D5     0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0     z5  
  D4     0 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0     z4  
  D3     0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1     z3  
  D2     0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 0     z2  
  D1     0 1 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1     z1  
  D0     0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 1     z0  
( det. =  -27 )
The order-64 cube constructed by this formula is complete and Nasik (mono-pan-1,2,3-agonal), and
the squared cube is a diagonal magic cube (bi-1,2,3-agonal).
The four triagonals of the cube are also trimagic (tri-3-agonal).
"Magic Cubes and Tesseracts" http://magcube.la.coocan.jp/magcube/en/
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