The question is a little vague but here is the answer to one possible interpretation of it. Let $G$ be a group given by finite presentation $R$. Suppose we know that $G$ is finite. Then there exists an algorithm to solve the word problem in $G$ (the word problem in every finite group is decidable), but there is no way we can know this algorithm explicitely, i.e. there is no algorithm which, given $R$ and the information that $G$ is finite, writes down an explicit Turing machine solving the word problem in $G$ (i.e. the uniform word problem in finite groups is undecidable, see Slobodskoĭ, A. M. Undecidability of the universal theory of finite groups. Algebra i Logika 20 (1981), no. 2, 207–230.)
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