Multimagic cube

In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. Thus, a magic cube is bimagic when it is 2-multimagic, and trimagic when it is 3-multimagic, tetramagic when it is 4-multimagic.[1] A P-multimagic cube is said to be semi-perfect if the k-th power cubes are perfect for 1 ≤ k < P, and the P-th power cube is semiperfect. If all P of the power cubes are perfect, the multimagic cube is said to be perfect.

The first known example of a bimagic cube was given by John Hendricks in 2000; it is a semiperfect cube of order 25 and magic constant 195325. In 2003, C. Bower discovered two semi-perfect bimagic cubes of order 16, and a perfect bimagic cube of order 32.[2]

MathWorld reports that only two trimagic cubes are known, discovered by C. Bower in 2003; a semiperfect cube of order 64 and a perfect cube of order 256.[3] It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.[4]

References

  1. ↑ Weisstein, Eric W. "Multimagic cube". MathWorld.
  2. ↑ Weisstein, Eric W. "Bimagic Cube". MathWorld.
  3. ↑ Weisstein, Eric W. "Trimagic Cube". MathWorld.
  4. ↑ Weisstein, Eric W. "Tetramagic Cube". MathWorld.

See also


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