S2T2

Jim Randell 5:21 pm on 20 December 2019 Permalink Reply
Tags: by: Graham Smithers ( 82 )   

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  Jim Randell 8:53 pm on 20 December 2019 Permalink | Reply

  This Python program determines the regular polygons with integer internal angles for 360 divisions (degrees) and 400 divisions (grads). It then finds the common values, and which of those are unique mod 10. It runs in 77ms.

  Run: [ @repl.it ]

  from enigma import (divisors_pairs, intersect, filter_unique, printf)
  
  # record n-gons by internal angles
  # t is the number of subdivisions in a circle
  # return map of: <internal angle> -> <number of sides>
  def ngons(t):
    r = dict()
    for (d, n) in divisors_pairs(t, every=1):
      if n > 2:
        r[t // 2 - d] = n
    return r
  
  # find n-gons for 360 subdivisions and 400 subdivisions
  A = ngons(360)
  B = ngons(400)
  
  # find interior angles that match
  ss = intersect([A.keys(), B.keys()])
  
  # filter them by the units digit
  ss = filter_unique(ss, (lambda x: x % 10))
  
  # output solution
  for i in ss.unique:
    printf("angle = {i} -> (a) {a}-gon, (b) {b}-gon", a=A[i], b=B[i])
  

  Solution: (a) Beth’s polygon has 72 sides; (b) Sam’s polygon has 16 sides.

  A 72-gon has internal angles measuring 175°. A 16-gon has internal angles measuring 157.5° = 175 grads.

  There are 4 numbers which will give regular n-gons for angles expressed in degrees and grads. These are:

  120: degrees → 6-gon (hexagon); grads → 5-gon (pentagon)
  150: degrees → 12-gon (dodecagon); grads → 8-gon (octagon)
  160: degrees → 18-gon (octadecagon); grads → 10-gon (decagon)
  175: degrees → 72-gon (heptacontakaidigon?); grads → 16-gon (hexadecagon)

  Obviously only the last of these is uniquely identified by the final (units) digit.

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• 

  GeoffR 8:53 am on 24 December 2019 Permalink | Reply

  This programme gives four answers to the constraints in the puzzle.
  The first three answers have a common units digit of zero, and the fourth answer has a unique digit of five, which provides the answer to the puzzle. Merry Xmas!

  % A Solution in MiniZinc
  include "globals.mzn";
  
  var 1..400: S;  % Sam's angle (grads)
  var 1..360: B;  % Beth's angle (degrees)
  
  var 1..150: n1; % edges to Sam's table mat
  var 1..120: n2; % edges to Beth's table mat
  constraint n1 != n2;
  
  % Internal angles equal gives 180 - 360/n1 = 200 - 400/n2 
  constraint B == 180 - 360 div n1 /\ 360 mod n1 == 0;
  constraint S == 200 - 400 div n2 /\ 400 mod n2 == 0;
  constraint 180 * n1 * n2 - 360 * n2 == 200 * n1 * n2 - 400 * n1;
   
  solve satisfy;
  
  output [" Edges for Sam's table mat = " ++ show(n1) ++ 
  ", Edges for Beth's table mat = " ++ show(n2)
  ++ "\n Sam's angle = " ++ show(S) ++ ", Beth's angle = " ++ show(B)];
  
  %  Edges for Sam's table mat = 6, Edges for Beth's table mat = 5
  %  Sam's angle = 120, Beth's angle = 120
  % ----------
  %  Edges for Sam's table mat = 12, Edges for Beth's table mat = 8
  %  Sam's angle = 150, Beth's angle = 150
  % ----------
  %  Edges for Sam's table mat = 18, Edges for Beth's table mat = 10
  %  Sam's angle = 160, Beth's angle = 160
  % ----------
  %  Edges for Sam's table mat = 72, Edges for Beth's table mat = 16
  %  Sam's angle = 175, Beth's angle = 175  << unique last digit
  % ----------
  % ==========
  
  
  

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