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$(a+b+c\cdots)\neq(a^{2}+b^{2}+c^{2}\cdots)$ given all distinct values for the variables?

When I came across this topic, it made me curious as to explore other possibilities, as here, what other two equations can XenoGraff use that have an equal number of variables, but have only one unique solution?

Anonym, who stated that this might be possible if it considered cubes, not squares, could be right, but if this isn't possible, then what are the possible other two equations that can be formed such that they have an equal number of variables with a unique solution?

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    Are there more? May I have some external re$f$erences on this topic? BTW is [this](http://www.jstor.org/stable/10.2307/2370317) and [this](http://www.jstor.org/stable/10.2307/1968085) link of any help?2012-05-08

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