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Consecutive Smith numbers. The first few Smith brothers are (728, 729), (2964, 2965), (3864, 3865), (4959, 4960), ... (OEIS A050219 and A050220).

Consider (1) If the probability distribution is governed by a Markov process, then P_3(y_1,t_1;y_2,t_2|y_3,t_3) = P_2(y_2,t_2|y_3,t_3) (2) = P_2(y_2|y_3,t_3-t_2). (3) ...

In a network with three graph edges at each graph vertex, the number of Hamiltonian cycles through a specified graph edge is 0 or even.

A surface parameterized in variables u and v is called smooth if the tangent vectors in the u and v directions satisfy T_uxT_v!=0, where AxB is a cross product.

A roll of two 1s (the lowest roll possible) on a pair of six-sided dice. The probability of rolling snake eyes is 1/36, or 2.777...%.

The 6-polyiamond illustrated above.

A surveying problem which asks: Determine the position of an unknown accessible point P by its bearings from three inaccessible known points A, B, and C.

A polyhedron with extra triangular faces, given by the Schläfli symbol s{p; q}.

Johnson solid J_(85).

The socle of a group G is the subgroup generated by its minimal normal subgroups. For example, the symmetric group S_4 has two nontrivial normal subgroups: A_4 and ...