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Re: Why does Solve give me no solutions for this in Version 8.0.1?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg119967] Re: Why does Solve give me no solutions for this in Version 8.0.1?
  • From: Adam Strzebonski <adams at wolfram.com>
  • Date: Sat, 2 Jul 2011 05:01:37 -0400 (EDT)

Solve gives only generic solutions, that is solutions that do not
require parameters to satisfy equations. In version 7 and earlier
Solve had an undocumented feature that equations which did not
syntactically contain variables were treated as "assumptions".
Hence, in version 7, the equation a == 0 in

In[1]:= Solve[x == 0 && a == 0, x]
Out[1]= {{x -> 0}}

was treated as an assumption and the solution x -> 0 was returned,
but the equivalent equation a + x == x in

In[2]:= Solve[x == 0 && a + x == x, x]
Out[2]= {}

was not treated as an assumption and no solution was returned.

In version 8 this feature was removed, so that equivalent systems
of equations give the same solutions.

In[1]:= Solve[x == 0 && a == 0, x]
Out[1]= {}

In[2]:= Solve[x == 0 && a + x == x, x]
Out[2]= {}

To obtain solutions that require equations on parameters, one can
use the option MaxExtraConditions.  With MaxExtraConditions -> n,
Solve includes solutions that require up to n equational conditions
on parameters.

In[3]:= Solve[x == 0 && a == 0, x, MaxExtraConditions -> 1]
Out[3]= {{x -> ConditionalExpression[0, a == 0]}}

In[4]:= Solve[x == 0 && a + x == x, x, MaxExtraConditions -> 1]
Out[4]= {{x -> ConditionalExpression[0, a == 0]}}

You can remove the ConditionalExpression wrapper using Normal.

In[5]:= Normal[%]
Out[5]= {{x -> 0}}

With MaxExtraConditions -> Automatic, Solve gives the solutions that
require the minimal number of equations on parameters (top-dimensional
part of the solution set).

In[7]:= eqns = {
    a + b + c + d + e + f + g + h + i + j + k +
    l + m + n + o + p + q + r + s - 190 == 0,
    a + b + c - 38 == 0,
    a + d + h - 38 == 0,
    a + e + j + o + s - 38 == 0,
    b + e + i + m - 38 == 0,
    b + f + k + p - 38 == 0,
    c + f + j + n + q - 38 == 0,
    c + g + l - 38 == 0,
    d + e + f + g - 38 == 0,
    d + i + n + r - 38 == 0,
    g + k + o + r - 38 == 0,
    h + i + j + k + l - 38 == 0,
    h + m + q - 38 == 0,
    l + p + s - 38 == 0,
    m + n + o + p - 38 == 0,
    q + r + s - 38 == 0
    }; 


In[8]:= Solve[eqns, b, MaxExtraConditions -> Automatic]//InputForm

Out[8]//InputForm=
{{b -> ConditionalExpression[j + n + o, q + r + s == 38 &&
      m + n + o + p == 38 && l + p + s == 38 && i + j + k + n + o + r == 
38 &&
      h - n - o - p - r - s == -38 && g + k + o + r == 38 &&
      f + j + k + n + o + p == 38 && e - k - n - o - p - r == -38 &&
      d - j - k - o == 0 && c - k - o - p - r - s == -38 &&
      a + j + k + n + 2*o + p + r + s == 76]}}

In[9]:= Normal[Solve[eqns, #, MaxExtraConditions -> Automatic]]&/@
{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s}//InputForm

Out[9]//InputForm=
{{{a -> 76 - j - k - n - 2*o - p - r - s}}, {{b -> j + n + o}},
  {{c -> -38 + k + o + p + r + s}}, {{d -> j + k + o}},
  {{e -> -38 + k + n + o + p + r}}, {{f -> 38 - j - k - n - o - p}},
  {{g -> 38 - k - o - r}}, {{h -> -38 + n + o + p + r + s}},
  {{i -> 38 - j - k - n - o - r}}, {{j -> 38 - i - k - n - o - r}},
  {{k -> 38 - i - j - n - o - r}}, {{l -> 38 - p - s}},
  {{m -> 38 - n - o - p}}, {{n -> 38 - m - o - p}}, {{o -> 38 - m - n - p}},
  {{p -> 38 - m - n - o}}, {{q -> 38 - r - s}}, {{r -> 38 - q - s}},
  {{s -> 38 - q - r}}}


Best regards,

Adam Strzebonski
Wolfram Research

-------- Original Message --------
Subject: [mg119967]     Why does Solve give me no solutions for this in 
Version 8.0.1?
Date:     Thu, 30 Jun 2011 20:40:24 -0400 (EDT)
From:     Phil J Taylor <xptaylor at gmail.com>
To:     mathgroup at smc.vnet.net



This system of equations for the Magic Hexagon is indeterminate.
  Solve still provides useful information in  Version 6.0

ClearAll[a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s];
eqns = {
    a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q + 
r + s
- 190 == 0,
    a + b + c - 38 == 0,
    a + d + h - 38 == 0,
    a + e + j + o + s - 38 == 0,
    b + e + i + m - 38 == 0,
    b + f + k + p - 38 == 0,
    c + f + j + n + q - 38 == 0,
    c + g + l - 38 == 0,
    d + e + f + g - 38 == 0,
    d + i + n + r - 38 == 0,
    g + k + o + r - 38 == 0,
    h + i + j + k + l - 38 == 0,
    h + m + q - 38 == 0,
    l + p + s - 38 == 0,
    m + n + o + p - 38 == 0,
    q + r + s - 38 == 0
    };

Join[
  Solve[eqns, b], Solve[eqns, d], Solve[eqns, g],
  Solve[eqns, m], Solve[eqns, p], Solve[eqns, j],
  Solve[eqns, r], Solve[eqns, e], Solve[eqns, f],
  Solve[eqns, i], Solve[eqns, k], Solve[eqns, n],
  Solve[eqns, o], Solve[eqns, a], Solve[eqns, c],
  Solve[eqns, h], Solve[eqns, l], Solve[eqns, q],
  Solve[eqns, s]]

Out[1]: {{b ->  j + n + o}, {d ->  j + k + o}, {g ->  i + j + n},
  {m ->  f + j + k}, {p ->  e + i + j}, {j ->  -38 + d + g + r},
  {r ->  -38 + h + l + m + p}, {e ->  -38 + h + k + q + r},
  {f ->  -38 + i + l + r + s}, {i ->  -38 + f + p + q + s},
  {k ->  -38 + e + m + q + s}, {n ->  -38 + g + h + k + l},
  {o ->  -38 + d + h + i + l}, {a ->  -38 + i + m + n + q + r},
  {c ->  -38 + k + o + p + r + s}, {h ->  -38 + n + o + p + r + s},
  {l ->  -38 + m + n + o + q +  r}, {q ->  -38 + g + k + l + o + p},
  {s ->  -38 + d + h + i + m +  n}}

Version 8.0.1 returns {}.



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