Upon rereading the problem, I find I hadn't been reading it correctly. I think I have it now, but if so I have a major difficulty with it.

A component p

_{i} of the vector pop is the numbers of eligible voters in state i. The hypothesis that all the v

_{i} are odd insures that there can't be a tie at the ballot box in any state. The corresponding component v

_{i} of the vote is the number of votes a candidate got in state i.

Therefore v

_{i} <= p

_{i}, and if 2*v

_{i} < p

_{i}, then the other candidate gets all that state's electors.

OK, the grand total number of electors is given to be 1001. Here is the difficulty I have with that: Both in the example with the non-conforming vector pop and erroneous computation with only 1000 electors being assigned, and in the vector pop in the puzzle itself,

**the number of electors is greater than the number of eligible voters!**