Multimagic cube: Difference between revisions
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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 [[exponentiation|powers]] for 1 | In [[mathematics]], a '''''P''-multimagic cube''' is a [[magic cube]] that remains magic even if all its numbers are replaced by their ''k''th [[exponentiation|powers]] for 1 ≤ ''k'' ≤ ''P''. {{nowrap|2-multimagic}} cubes are called '''bimagic''', {{nowrap|3-multimagic}} cubes are called '''trimagic''', and {{nowrap|4-multimagic}} cubes '''tetramagic'''.<ref name=Multi>{{MathWorld|id=MultimagicCube}}</ref> A {{nowrap|''P''-multimagic}} cube is said to be '''semi-perfect''' if the ''k''th power cubes are [[perfect magic cube|perfect]] for 1 ≤ ''k'' < ''P'', and the ''P''th power cube is [[semiperfect magic cube|semiperfect]]. If all ''P'' of the power cubes are perfect, the {{nowrap|''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 magic cube|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.<ref name=Bi>{{MathWorld|title=Bimagic Cube|id=BimagicCube}}</ref> | The first known example of a bimagic cube was given by [[John Hendricks]] in 2000; it is a [[semiperfect magic cube|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.<ref name=Bi>{{MathWorld|title=Bimagic Cube|id=BimagicCube}}</ref> | ||
[[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.<ref name=Tri>{{MathWorld|id=TriMagic|title=Trimagic Cube}}</ref> It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.<ref name=Tetra>{{MathWorld|title=Tetramagic Cube|id=TetramagicCube}}</ref> | [[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.<ref name=Tri>{{MathWorld|id=TriMagic|title=Trimagic Cube}}</ref> It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.<ref name=Tetra>{{MathWorld|title=Tetramagic Cube|id=TetramagicCube}}</ref> | ||
In 2011, Emlyn Ellis Addison found a mod-9 symmetric semiperfect tetramagic cube of order 9, intended as a methodology for structuring musical compositions.<ref>{{citation|url=https://emlynellisaddison.com/tetramagic_cube/Mod-9%20Symmetric%20Semiperfect%20Tetramagic%20Cube.pdf|title=The Numerical Model Behind Empathy Alpha|work=emlynellisaddison.com|format=PDF|first=Emlyn Ellis|last=Addison|date=January 1, 2022|access-date=2025-09-21}}</ref> | |||
== References == | == References == | ||
Latest revision as of 18:12, 15 December 2025
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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 kth 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 kth power cubes are perfect for 1 ≤ k < P, and the Pth 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]
References
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