New Scientist Enigma 1354: Sound idea

     Here is a solution to Enigma 1354 which appeared recently in Enigmatic Code

The problem is this:

Joe’s daughter has been asking him for weeks to make a wind chime. So this week Joe cut a length of stainless steel tubing into 10 lengths from 1 cm to 10 cm in steps of 1 cm. He also cut out a disc from a sheet of stainless steel and drilled 10 evenly spaced holes round the perimeter and one in the centre.

Joe hung one tube from each hole and hung the disc up by a thread from its centre. That was a big mistake as the disc tilted right over.

So Joe rearranged the tubes until the disc balanced perfectly. In making a note of the lengths of the tubes in order round the disc the result was an 11-digit number.

What is the smallest of all the 11-digit numbers Joe could have written down?

In order for the wind chime to balance, moments about the axis af must balance

$ (b+e-g-j)sin(\frac{π}{5}) + (c+d-h-i)sin(\frac{2π}{5}) = 0 $ ..................... (1)

Noting that $sin( \frac{2π}{5}) = 2 sin(\frac{π}{5}) cos(\frac{π}{5})$ , dividing (1) through by $sin(\frac{π}{5})$

$ (b+e-g-j) +2 (c+d-h-i)cos(\frac{π}{5}) = 0 $ ..................... (2)

Substituting $ cos(\frac{π}{5}) = \frac{(1+\sqrt{5})}{4} $ into (2)

$ b+e-g-j +\frac{c+d-h-i}{2} +\frac{(c+d-h-i)\sqrt{5}}{2} = 0$

As a,b,...,i,j are all integers,$ c+d-h-i = 0$ and $b+e-g-j +\frac{c+d-h-i}{2}     =0$

So $c+d=h+i$ and $b+e=g+j$

So knowing five adjacent values, say, a,b,c,d,e, allows the other five to be deduced:

$f = 55 - a - (b+c+d+e) - (g+h+i+j) = 55-a-2(b+c+d+e)$

And so on round the circle.

So, some code:

from itertools import permutations as perm

a=10
onetoten=frozenset(range(1,11))

for b,c,d,e in perm(range(1,10),4):
  f=55-a-2*(b+c+d+e)
  g=55-b-2*(c+d+e+f)
  h=55-c-2*(d+e+f+g)
  i=55-d-2*(e+f+g+h)
  j=55-e-2*(f+g+h+i)

  s=(a,b,c,d,e,f,g,h,i,j)
  if set(s)==onetoten:

    # sanity check
    if all( sum(s[i%10] for i in xrange(m,m+4))
            ==sum(s[i%10] for i in xrange(m+5,m+9))
            for m in xrange(10)):
      print "solution", a,b,c,d,e,f,g,h,i,j
      break