# Multimagic cube

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This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Multimagic cube" – news · newspapers · books · scholar · JSTOR (October 2011)

In [mathematics](/source/Mathematics), a ***P*-multimagic cube** is a [magic cube](/source/Magic_cube) that remains magic even if all its numbers are replaced by their *k*th [powers](/source/Exponentiation) 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](/source/Perfect_magic_cube) for 1 ≤ *k* < *P*, and the *P*th power cube is [semiperfect](/source/Semiperfect_magic_cube). 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](/source/John_Hendricks) in 2000; it is a [semiperfect](/source/Semiperfect_magic_cube) cube of order 25 and [magic constant](/source/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](/source/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]

## References

1. **[^](#cite_ref-Multi_1-0)** [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Multimagic cube"](https://mathworld.wolfram.com/MultimagicCube.html). *[MathWorld](/source/MathWorld)*.

1. **[^](#cite_ref-Bi_2-0)** [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Bimagic Cube"](https://mathworld.wolfram.com/BimagicCube.html). *[MathWorld](/source/MathWorld)*.

1. **[^](#cite_ref-Tri_3-0)** [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Trimagic Cube"](https://mathworld.wolfram.com/TriMagic.html). *[MathWorld](/source/MathWorld)*.

1. **[^](#cite_ref-Tetra_4-0)** [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Tetramagic Cube"](https://mathworld.wolfram.com/TetramagicCube.html). *[MathWorld](/source/MathWorld)*.

1. **[^](#cite_ref-5)** Addison, Emlyn Ellis (January 1, 2022), ["The Numerical Model Behind Empathy Alpha"](https://emlynellisaddison.com/tetramagic_cube/Mod-9%20Symmetric%20Semiperfect%20Tetramagic%20Cube.pdf) (PDF), *emlynellisaddison.com*, retrieved 2025-09-21

## See also

- [Magic square](/source/Magic_square)

- [Multimagic square](/source/Multimagic_square)

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