WASHINGTON -- Were it not for the 12th Amendment, ratified on June 15, 1804, presidential nominees could, and probably would prefer to, run alone, without being saddled with pesky vice presidential running mates, who can be embarrassments. Unfortunately, the presidential election of 1800 happened.

It was "a magnificent catastrophe" (that is the title of a splendid new book on it, by Edward J. Larson of Pepperdine University). As the two-party system, unanticipated by the Constitution's otherwise farsighted Framers, was crystallizing, Thomas Jefferson and Aaron Burr each received the same number of electoral votes. This made a hash of the Framers' plan for electing presidents without having vice presidential candidates. Under their plan, which worked fine for three elections, the presidential candidate with the second highest number of electoral votes -- John Adams twice, then Jefferson -- became vice president.

It took the House of Representatives until Feb. 17, 1801, and 36 ballots, to break the Jefferson-Burr tie and make Jefferson president. Which is why, before the 1804 election, the 12th Amendment was written to require that electoral votes shall be cast "for president and vice president" with the expectation that the candidates would run on a party ticket and why Barack Obama or Hillary Clinton will need a running mate.

They both might need the same white man. Or perhaps Obama will need a different one. In any case, consider the probable logic of their choices.

Presidential politics, like football, is simple in objective but complex in execution. The game is Get to 270 -- electoral votes, that is. A person who captures a presidential nomination gives (sometimes perfunctory) thought to whether or not this or that potential running mate might make a crackerjack president (John Nance Garner? Henry Wallace? Alben Barkley? John Sparkman? Estes Kefauver? Bill Miller? Spiro Agnew?). Then the presidential nominee gives serious thought to whether this or that running mate might contribute a winning edge in a closely contested state with a significant number of electoral votes. So, speaking of Ohio. ...

In 2000, George W. Bush beat Al Gore 271-266 in electoral votes. Had Bush not carried Ohio, which he did with 50 percent of the popular vote, he would have lost. In 2004, he beat Kerry 286-251. Without Ohio's 20 electoral votes, which Bush won 51-49, Kerry would be president. By various scandals and ineptitudes, Ohio's Republican Party made such a mess that in 2006 Sherrod Brown, a congressman, defeated an incumbent Republican senator, Mike DeWine, and another congressman, Ted Strickland, became governor.

Obama needs a running mate who is older than he is (46), who is not in Washington, who has executive experience and national security experience, who comes from a swing state and from a demographic cohort that Democratic presidential candidates have lost in every election since 1964 -- white males. Strickland, who will be 67 in August, is the son of a steelworker and the first Democrat elected to Ohio's governorship since 1986. He is an ordained Methodist minister who was born and raised in, and represented in Congress, conservative southeastern Appalachian Ohio. He has everything Obama needs -- even an A rating from the National Rifle Association -- except national security experience.

Clinton, too, needs Ohio, and owes Strickland, who helped her win in the primary. In politics, gratitude is optional, but a pleasant surprise.

Where would McCain make up for losing Ohio's electoral votes? Perhaps he could turn New Hampshire back to red: He has twice won its primary, and it was one of only three states -- the others were Iowa and New Mexico -- which changed partisan alignment between 2000 and 2004 (when Kerry won with 50 percent). But New Hampshire has just four electoral votes. Perhaps another 16 can be found somewhere, but going into the fall with Ohio out of reach would put McCain in a deep hole.

If, however, McCain, whose strongest feelings and only competence concern national security, succeeds in making this a national security election -- if he does not, he almost certainly loses -- Obama could consider a different running mate: Sam Nunn, 69, the former four-term senator from Georgia. Winning the state's 15 electoral votes in November would be difficult even with Nunn on the ticket. Still, the value of running with Nunn, who chaired the Senate Armed Services Committee, would be substantial.

But not more substantial than the benefits from putting Ohio, in August, out of McCain's November reach. That would free Obama's or Clinton's time and money for use elsewhere and compel McCain to wager time and money on other, and problematic, states. The choice of Strickland would be, in military terminology, a force multiplier.