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A set of n magic circles is a numbering of the intersections of the n circles such that the sum over all intersections is the same constant for all circles. The above sets of ...

The pair group of a group G is the group that acts on the 2-subsets of {1,...,p} whose permutations are induced by G. Pair groups can be calculated using PairGroup[g] in the ...

A planted plane tree (V,E,v,alpha) is defined as a vertex set V, edges set E, root v, and order relation alpha on V which satisfies 1. For x,y in V if rho(x)<rho(y), then ...

A planted tree is a rooted tree whose root vertex has vertex degree 1. The number of planted trees of n nodes is T_(n-1), where T_(n-1) is the number of rooted trees of n-1 ...

A simple polyhedron, also called a simplicial polyhedron, is a polyhedron that is topologically equivalent to a sphere (i.e., if it were inflated, it would produce a sphere) ...

A strongly binary tree is a rooted tree for which the root is adjacent to either zero or two vertices, and all non-root vertices are adjacent to either one or three vertices ...

A trivalent tree, also called a 3-valent tree or a 3-Cayley tree, is a tree for which each node has vertex degree <=3. The numbers of trivalent trees on n=1, 2, ... nodes are ...

In a given circle, find an isosceles triangle whose legs pass through two given points inside the circle. This can be restated as: from two points in the plane of a circle, ...

There are a number of attractive polyhedron compounds of two cubes. The first (left figures) is obtained by allowing two cubes to share opposite polyhedron vertices then ...

The path traced out by a point P on the edge of a circle of radius b rolling on the outside of a circle of radius a. An epicycloid is therefore an epitrochoid with h=b. ...