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

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  Re: Why does Solve give me no solutions for this in Version 8.0.1? Daniel Lichtblau <danl@wolfram.com> - 2011-07-02 09:06 +0000

#3455 — Re: Why does Solve give me no solutions for this in Version 8.0.1?

On 06/30/2011 07:40 PM, Phil J Taylor wrote:
> 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 {}.


There are no generic solutions: each separate system forces equations 
involving non-Solve variables. See
Documentation Center > Solve > Options > MaxExtraConditions

You could do e.g.

In[563]:= Solve[eqns, b, MaxExtraConditions -> Infinity]

Out[563]= {{b ->
    ConditionalExpression[j + n + o,
     d - j - k - o == 0 && f + j + k + n + o + p == 38 &&
      m + n + o + p == 38 && e - k - n - o - p - r == -38 &&
      g + k + o + r == 38 && i + j + k + n + o + r == 38 &&
      c - k - o - p - r - s == -38 && h - n - o - p - r - s == -38 &&
      l + p + s == 38 && a + j + k + n + 2 o + p + r + s == 76 &&
      q + r + s == 38]}}

I'd suggest instead just doing

Solve[eqns]

An addition to being simpler to input, you will not have a "solution 
set" where r is in terms of l, and l is in terms of r.

If you want to specify a set of variables to solve for in terms of all 
the rest, that shopuld be fine too. I'd make sure it is large enough 
that there are no remaining relations in terms of non-specified 
variables (or again you'll get {}). Alternatively you could set 
MaxExtraConditions to some high value.


Daniel Lichtblau
Wolfram Research

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