Magic Buckyball

In geometry, the truncated icosahedron is an Archimedean solid with 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. The football (soccer ball in the USA) has the shape of a truncated icosahedron.

In 1985 a new form of carbon was reported. (Kroto, H. W.; Heath, J. R.; O'Brien, S. C.; Curl, R. F.; Smalley, R. E. (1985). "C60: Buckminsterfullerene". Nature 318 (6042): 162–163.) The molecule they discovered was a spherical cage of 60 carbon atoms, C60. They called their new discovery Buckminsterfullerene in honor of Buckminster Fuller the builder and promoter of the geodesic dome which is a similar shaped structure. The geodesic dome was their inspiration for determining the structure of their discovery.

The carbon atoms of Buckminsterfullerene are arranged as the vertices of a truncated icosahedron. The edges of the truncated icosahedron are carbon to carbon bonds. The name is frequently shortened to buckyball

In 1996 Robert Curl, Harold Kroto and Richard Smalley received the Nobel prize in Chemistry for their 1985 discovery of buckyballs. Buckyballs have recently been discovered in planetary nebulae.

It is possible to make a magic buckyball using the polygon faces on the surface of the buckyball. The 32 faces of the buckyball below are numbered with sequential numbers from 1 to 32. Every pentagon plus the surrounding five hexagons add to the same sum making it a magic buckyball.

With different arrangements of the numbers 1-32 it is possible to make magic buckyballs with nine different sums; 79, 84, 89, 94, 99, 104, 109, 114, and 119. In the magic buckyball below all nine sums can be chosen and an example of a magic buckyball with the correct sum will be shown.

To make a magic buckyball with a sum of 79 the numbers 21-32 must be used in the pentagons and the numbers 1-20 must be used in the hexagons. The pentagons and hexagons for a magic buckyball with the sum of 79 must use only these number ranges.

The magic buckyball with the sum of 119 is the complement of the one with a sum of 79. The complement for a number, n, in the magic buckyball is 33 - n. In this magic buckyball the pentagons contain the numbers 1-12 and the hexagons contain the numbers 13-32. The pentagons and hexagons for a magic buckyball with the sum of 119 must use only these number ranges.

For the other sums the number ranges are not as restricted. Numerous number combinations for the pentagons and hexagons are possible.

The magic buckyballs shown below with the sums 84 and 114 are complements as are 89 and 109. The magic buckyballs with sums 94 and 104 are also complements but they have an additional feature. The pentagons use only even numbers when the sum is 94 and only odd numbers when the sum is 104.

The magic buckyball that sums to 99 is self-complementary. Opposite side of this buckyball are complements. and will add to 33. Self-complementary buckyballs are only possible with the sum of 99. This is because the average value for both the pentagons and hexagons is 15.5, the same as the average value of the numbers 1 to 32.

Select a sum

The sum of any pentagon and the surrounding five hexagons is: