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There are 432 enantiomorphous and 415 chiral fully supported stellations of the icosidodecahedron. Using Miller's rules gives 7071672 enantiomorphous stellations and an ...

If p divides the numerator of the Bernoulli number B_(2k) for 0<2k<p-1, then (p,2k) is called an irregular pair. For p<30000, the irregular pairs of various forms are p=16843 ...

Applying the Kaprekar routine to 4-digit number reaches 0 for exactly 77 4-digit numbers, while the remainder give 6174 in at most 8 iterations. The value 6174 is sometimes ...

The Krohn-Rhodes complexity, also called the group complexity or simply "the complexity," of a finite semigroup S is the smallest number of groups in a wreath product of ...

Let a graph G have exactly 2n-3 graph edges, where n is the number of graph vertices in G. Then G is "generically" rigid in R^2 iff e^'<=2n^'-3 for every subgraph of G having ...

A longimeter is a transparent sheet of plastic with a regular grid of lines inclined at an angle of 30 degrees to the sides of the sheet. By counting the number of squares ...

A magic hexagram is a hexagram partitioned into triangles such that the sums of numbers in the six directions illustrated above sum to the same number. There are exactly two ...

The matrix tree theorem, also called Kirchhoff's matrix-tree theorem (Buekenhout and Parker 1998), states that the number of nonidentical spanning trees of a graph G is equal ...

A large number defined as where the circle notation denotes "n in n squares," and triangles and squares are expanded in terms of Steinhaus-Moser notation (Steinhaus 1999, pp. ...

If the period of a repeating decimal for a/p, where p is prime and a/p is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves ...