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I'm looking for a way to efficiently determine the solutions ( i.e. sets of $x_1^2, x_2^2, ..., x_k^2 $ ) that satisfy     $$n^2 = x_1^2 + x_2^2 + x_3^2 + ... + x_k^2 $$ $$x_1, x_2, ..., x_k, n, k \in\mathbb{Z}_{>0}$$

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    I don't understand why the author says that 21 is the lowest possible order, I can dissect $11^2$ into $10^2+4^2+2^2+1^2$, I'm obviously missing something here, also, I cannot see an approach for this in the link I provided. – 2017-02-11
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    The Lagrange theorem about four squares states that it can be done for sure with $k \leq 4$. So I looked up it in wikipedia, where it is said that some people Rabin and Shallit, have written an algorithm for find that four. The link brought me here http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160390713/abstract;jsessionid=7E409C3AC70B2CC660D0A1E2184C7AD9.f02t04 . Maybe you could download it from somewhere else. – 2017-02-11
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    What you did was purely algebraic and ignores the geometric aspect of the dissection. Surely you can see that fitting an $10\times10$ square inside an $11\times11$ square leaves no room for any square that is not $1\times1$. – 2017-02-11
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    @AbudaDumiaty What you're missing is in the picture linked in the question. The OP wants a geometric dissection of a square. – 2017-02-11
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    I may have figured out the difference between my example and the one in the link, with mine it's not possible to arrange the four squares (side lengths: 10, 4, 2, 1) to form the square $11^2$, which is fine, as this is still what I'm looking for, even if it cannot be demonstrated visually. EDIT: @Fimpellizieri, yeah thanks, I just realized the difference. – 2017-02-11
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    @AbudaDumiaty, if you don't want geometric decomposition like in the link above, then would you consider restricting you problem to $k=4$ since it is possible to do for any integer? Or would you like $k$ to be as small as possible? As large as possible? – 2017-02-11
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    @YuriyS, I would like to allow $k$ to be any value, no matter how large or small it is. Specifically, I'm interested in finding all possible solutions (i.e. sets of $x1, x2, ..., xk$) for a given $n$. – 2017-02-11
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    @AbudaDumiaty, ok. You should edit the question, because what you actually want has nothing to do with dissecting squares and with your link – 2017-02-11

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