Let $n,k$ be positive integers. What is the smallest value of $N$ such that for any $N$ vectors (may be repeated) in $(\mathbb Z/(n))^k$, one can pick $n$ vectors whose sum is $0$? 


My guess is $N=2^k(n-1)+1$. It is certainly sharp: one can pick our set to be $n-1$ copies of the set $(a\_1,...,a\_k)$, with each $a\_i=0$ or $1$. The case $k=1$ is some math competition question (I think, but can't remember the exact reference). Does anyone know of some references? Thanks. 




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Thank you all! I wish I could accept all the answers, they are very helpful!



Sum of $n$ vectors in $(\mathbb Z/n)^k$
Your guess is correct for k=1 and 2, but when k is bigger, things get more complicated. For instance, when k=n=3, N=19. For a summary of some known results, see:


[Elsholtz, C. Lower Bounds For Multidimensional Zero Sums. *Combinatorica* **24**, 351–358 (2004).](https://doi.org/10.1007/s00493-004-0022-y)


