Multimagic cube (original) (raw)

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In mathematics, a _P_-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their _k_th powers for 1 ≤ k ≤ P. 2-multimagic cubes are called bimagic, 3-multimagic cubes are called trimagic, and 4-multimagic cubes tetramagic.[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 _P_-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]

In 2011, Emlyn Ellis Addison found a mod-9 symmetric semiperfect tetramagic cube of order 9, intended as a methodology for structuring musical compositions.[5]

  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.
  5. ^ Addison, Emlyn Ellis (January 1, 2022), "The Numerical Model Behind Empathy Alpha" (PDF), emlynellisaddison.com, retrieved 2025-09-21