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 {}.