Some problems in Number Theory

Abstract: 

I'll discuss some problems that have bothered me for a while on small sums of roots of unity, on covering congruences, and on semi-magic squares. Among the questions I may discuss are:How small can a nonzero sum of five nnth roots of unity be?

1. How small can a nonzero sum of five nnth roots of unity be?
2. Can Z⊕ZZ⊕Z be written as a finite, disjoint union of cosets of distinct, proper subgroups?
3. What's the ``smallest'' 4×44×4 matrix with natural number entries and all row and column sums equal that can't be written as an integer linear combination of nine permutation matrices?