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Physical Sciences and Mathematics Commons™
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Full-Text Articles in Physical Sciences and Mathematics
Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin
Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin
All HMC Faculty Publications and Research
We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.
A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger
A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger
All HMC Faculty Publications and Research
We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all p ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.