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Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$.

Find the minimum number of solution of the system

$$P_{2i} + P_{2i+1} = \lambda_i, \forall i\leq k$$

with the condition that the $P_1, ..., P_{2k+1}$ are pairwise distinct.

The law "+" on the $n-$bits strings is the "xor" ie we consider the n-bit strings as element of $(\mathbb{Z}/2\mathbb{Z})^n$.

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How about fixing the typo(s) and motivating your question to convince the community that this is not a homework problem? –  Seva Feb 11 '13 at 14:35
    
I think this is just a fragment of a question -- significant parts of the text are missing. –  Stefan Kohl Feb 11 '13 at 19:42

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