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According to the documentation, FindInstance[] return the empty set when it does not find a solution and one of the examples claims that this is proof that no solution exists. So if FindInstance[] returns unevaluated, does this mean that Mathematica could not decide of there was a solution or not? This behavior does not seem to be documented.

Example:

s = FindInstance[a^2 + b^2 + c^2 == k && d^2 + e^2 + f^2 == k && g^2 + h^2 + i^2 == k && a^2 + d^2 + g^2 == k && b^2 + e^2 + h^2 == k && c^2 + f^2 + i^2 == k && a^2 + e^2 + i^2 == k && c^2 + e^2 + g^2 == k && a > 0 && b > 0 && c > 0 && d > 0 && e > 0 && f > 0 && g > 0 &&  h > 0 && i > 0 && k > 0 && a != b && a != c && a != d && a != e && a != f && a != g && a != h && a != i && b != c && b != d && b != e && b != f && b != g && b != h && b != i && c != d && c != e && c != f && c != g && c != h && c != i && d != e && d != e && d != f && d != g && d != h && d != i && e != f &&e != g && e != h && e != i && f != g && f != h && f != i && g != h && g != i && h != i, {a, b, c, d, e, f, g, h, i, k},Integers]
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  • $\begingroup$ Please provide an example. $\endgroup$ – bbgodfrey Apr 18 '15 at 15:08
  • $\begingroup$ You might try MaxMemoryUsed[FindInstance[...]] and check the relation to your system's RAM. $\endgroup$ – Jinxed Apr 18 '15 at 15:55
  • $\begingroup$ Generally speaking, Mathematica functions return unevaluated if the input is insufficient to compute an answer (e.g. Sin[x]). Often they issue an error or warning if there is a recognizable problem with the input. FindInstance usually issues FindInstance::nsmet when it knows it does not know how to find solutions to the equation. That suggests there is some particular issue with your problem that might be specific to it. Or perhaps someone on this site has run into this before and can share what was wrong with their problem. If you're lucky, it might be the same. $\endgroup$ – Michael E2 Apr 18 '15 at 16:27

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