What if Stephen Douglas instead of Abraham Lincoln had won the U.S. presidential election of 1860? What if John F. Kennedy had not carried some of the eight states he won by 2 percentage points or fewer in 1960? What if six hundred more people in Florida had voted for Al Gore in 2000? And what if, in that same year, the U.S. House of Representatives had more than 435 members? In any close or contested election, allegations of miscounting, fraud, or other forms of conspiracy arise. The purpose of this article is not to consider such issues. Instead, it looks at a factor that always plays a role in U.S. presidential elections and that in close elections plays a pivotal role: the apportionment of the U.S. House of Representatives. The connection between apportionment and elections--a connection that combines mathematics, history, and politics--is a great way to motivate students. Apportionment is a mathematically intriguing question that students can explore in many ways. (Contains 4 tables.)