```Date: Jan 14, 2001 2:27 PM
Author: Kirby Urner
Subject: Pool Hall Math

POOL HALL MATH             Oregon Curriculum NetworkYou can start with 15 pool balls (bring 'em to class)to kick off a discussion of Triangular Numbers.Links to Gauss:  Tn = 1 + 2 + 3... + nWhich he solved (age 7) by going:   1     2     3 ...   n   n   n-1   n-2 ...   1  ---------------------- n+1   n+1   n+1 ... n+1  = n(n+1) which is twice the sum he wanted so: Tn = n(n+1)/2Then stack smaller triangles on the 15 to an apexball (good if you've got the pool balls, need 35 in all -- switch to ping pong balls if pool ballstoo spendy).  That's a tetrahedron.  If you're lucky enough to have a CLI, you can do things like:  >>> def tri(n): return n*(n+1)/2  >>> def tetra(n):	 if n==1: return 1	 return tri(n) + tetra(n-1)  >>> tetra(5)  35Now if you're dealing with somewhat older students,it's useful to say "as you know, just as area increasesas a 2nd power of linear increase, so does volume increase as a 3rd power, so we'd expect to find anexpression in the 3rd degree for the Tetrahedral Number Ttn, just as Gauss did for Tn".The solution involves writing:    A n^3 + B n^2 + C = Ttnand then substituting known pairs (n, Ttn), to get 3 linear equations in 3 unknowns.  Could go with matrix methods if that's what's up in class.KirbyLinks:Ttn to CCP (cube centric packing = fcc = ivm)CCP and Alexander Graham Bell (kites + truss)CCP and R. Buckminster Fuller (octet truss)Sequences Tn and Ttn in Pascal's Triangle (Math Forum)Pascal's Triangle as itself figurate (a triangle)Pascal's Triangle and Binomial TheormPascal's Tetrahedron (advanced: trinomial theorem)
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