function a = archimedes ( )

%*****************************************************************************80
%
%% ARCHIMEDES returns the 7 by 8 Archimedes matrix.
%
%  Formula:
%
%      6   -5    0   -6    0    0    0    0
%      0   20   -9  -20    0    0    0    0
%    -13    0   42  -42    0    0    0    0
%      0   -7    0    0   12   -7    0    0
%      0    0   -9    0    0   20   -9    0
%      0    0    0  -11    0    0   30  -11
%    -13    0    0    0  -13    0    0   42
%
%  Discussion:
%
%    "The sun god had a herd of cattle, consisting of bulls and cows,
%    one part of which was white, a second black, a third spotted, and
%    a fourth brown.  Among the bulls, the number of white ones was
%    one half plus one third the number of the black greater than
%    the brown; the number of the black, one quarter plus one fifth
%    the number of the spotted greater than the brown; the number of
%    the spotted, one sixth and one seventh the number of the white
%    greater than the brown.  Among the cows, the number of white ones
%    was one third plus one quarter of the total black cattle; the number
%    of the black, one quarter plus one fifth the total of the spotted
%    cattle; the number of spotted, one fifth plus one sixth the total
%    of the brown cattle; the number of the brown, one sixth plus one
%    seventh the total of the white cattle.  What was the composition
%    of the herd?"
%
%    The 7 relations involving the 8 numbers W, X, Y, Z, w, x, y, z,
%    have the form:
%
%      W = ( 5/ 6) *   X + Z
%      X = ( 9/20) *   Y + Z
%      Y = (13/42) *   W + Z
%      w = ( 7/12) * ( X + x )
%      x = ( 9/20) * ( Y + y )
%      y = (11/30) * ( Z + z )
%      z = (13/42) * ( W + w )
%
%    These equations may be multiplied through to an integral form
%    that is embodied in the above matrix.
%
%    A more complicated second part of the problem imposes additional
%    constraints (W+X must be square, Y+Z must be a triangular number).
%
%  Properties:
%
%    A is integral: int ( A ) = A.
%
%    A has full row rank.
%
%    It is desired to know a solution X in positive integers of
%
%      A * X = 0.
%
%    The null space of A is spanned by multiples of the null vector:
%
%      [ 10,366,482 ]
%      [  7,460,514 ]
%      [  7,358,060 ]
%      [  4,149,387 ]
%      [  7,206,360 ]
%      [  4,893,246 ]
%      [  3,515,820 ]
%      [  5,439,213 ]
%
%    and this is the smallest positive integer solution of the
%    equation A * X = 0.
%
%    Thus, for the "simple" part of Archimedes's problem, the total number
%    of cattle of the Sun is 50,389,082.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    27 September 2007
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Eric Weisstein,
%    CRC Concise Encyclopedia of Mathematics,
%    CRC Press, 1998.
%
%  Parameters:
%
%    Output, integer A(7,8), the matrix.
%
  a(1:7,1:8) = [ ...
     6, -5,  0,  -6,   0,  0,  0,   0; ...
     0, 20, -9, -20,   0,  0,  0,   0; ...
   -13,  0, 42, -42,   0,  0,  0,   0; ...
     0, -7,  0,   0,  12, -7,  0,   0; ...
     0,  0, -9,   0,   0, 20, -9,   0; ...
     0,  0,  0, -11,   0,  0, 30, -11; ...
   -13,  0,  0,   0, -13,  0,  0,  42 ];

  return
end