Pascal's triangle Construct a "d"-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to P(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P("x"), have a total of "x" dots composing the target shape. P("x") then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P("x") = "1" and P("x") = "x", which is the sequence of natural numbers. The number of dots in each layer corresponds
Circle packing in an equilateral triangle Circle packing in an equilateral triangle Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack "n" unit circles into the smallest possible equilateral triangle. Optimal solutions are known for "n" < 13 and for any triangular number of circles, and conjectures are available for "n" < 28. A conjecture of Paul Erdős and Norman Oler states that, if is a triangular number, then the optimal packings of and of circles have the same side length: that is, according to the conjecture, an optimal packing for circles can be found by