S2T2

Jim Randell 8:38 am on 3 June 2021 Permalink Reply
Tags: by: Victor Bryant ( 139 )   

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  Jim Randell 8:38 am on 3 June 2021 Permalink | Reply

  (See also: Enigma 1296).

  The number of upward pointing triangles on a triangular grid of size n is the sum of the triangular numbers from 1 to n. These are the tetrahedral (or triangular pyramidal) numbers (see: OEIS A000292).

  The following Python program runs in 50ms.

  Run: [ @replit ]

  from enigma import (irange, inf, tri, mlcm, call, printf)
  
  # generate increasing (grid size, upward triangles) pairs
  def generate(m=inf):
    t = 0
    for n in irange(1, m):
      t += tri(n)
      yield (n, t)
  
  # find a solution divisible by d
  d = call(mlcm, irange(1, 9))
  for (n, t) in generate():
    if t % d == 0:
      printf("n={n} t={t}")
      break  # we only need the first solution
  

  Solution: Trix is 54.

  ---

  Using the formula:

  ups(n) = n(n + 1)(n + 2) / 6

  We are looking for solutions where ups(n) is a multiple of 2520:

  n(n + 1)(n + 2) / 6 = 2520k
  n(n + 1)(n + 2) = 15120k

  Which gives the following program:

  from enigma import (irange, inf, tuples, multiply, printf)
  
  for ns in tuples(irange(1, inf), 3):
    t = multiply(ns)
    if t % 15120 == 0:
      printf("n={n} t={t}", n=ns[0])
      break
  

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