10/16/2012 08:12 pm ET Updated Dec 16, 2012

U.S. Presidential Election Paradox


Question: What is the fewest number of votes with which you could be elected President of the United States? Say that there are just two candidates and that half the population in each state votes. Hint: You can win with fewer than half the votes.

Answer. Technically you just need to win about half the votes in states with about half the electoral votes, or about 25 percent of the popular vote. Actually, since smaller states get more electoral votes per resident, you can win with 39 small states plus the District of Columbia and under 22 percent of the popular vote. So Romney could beat Obama handily if he could just redistribute his votes among the smaller states. In 1888, Benjamin Harrison beat Grover Cleveland with fewer popular votes by winning lots of states by a few votes and losing the rest of the states big.

If you consider the different voter turnout rates in different states, you could win with still fewer votes by winning states with low turnouts. In 1992, voter turnout varied from 42 percent of the voting-age population in Hawaii to over 70 percent in Maine, Minnesota, and Montana. A recent article in Math Horizons by Chuck Wessell says you could win with under 18 percent of the vote.

Question. What if there were three candidates?

Answer. With three candidates, a different argument says you need less than 0.1 percent of the vote, just enough to win Wyoming. If your opponents each fail to win a majority of electoral votes, the Constitution provides for the House of Representatives to choose among the three top electoral vote winners, including you, and if you can get the votes of 26 small state delegations in the House, you are elected! In 1824, Andrew Jackson won in popular and electoral votes, but failed to get a majority, and the House elected John Quincy Adams.

For such reasons many recommend that the Electoral College voting system be replaced by election by popular vote. Popular vote not only would be fairer, but also would encourage the candidates to campaign in the whole country, instead of focusing on a few battleground states.

Based on a chapter from "Math Chat Book."