Source: J. Witkowski's "Math Modeling and Presidential Elections" *Mathematics Teacher* Oct 1992

**Hint:** You will need an ordered list of the composition of the House of Representatives (e.g. 7 states have 1 Representative, 6 have 2, etc.).

If the "elected" candidate wins w states, then we have the sum of electoral votes for that candidate being: (r_{1}+2) + (r_{2}+2) +...+ (r_{w}+2) > 269. You need to solve this equation...!

**Solution Commentary:** Performing some algebra on hint's equation, we have: r_{1} + r_{2} +...+ r_{w} > 269 - 2w. Now, as you select states (with r Representatives) for this equation, you want to minimize r_{1} + r_{2} +...+ r_{w} but also note that as the number of states w increases, the expression 269 - 2w decreases.

Using your ordered list, start with states with small populations first (e.g. with r = 1, w = 7+D.C. = 8, r_{1} + r_{2} +...+ r_{w} = 8 and 269-2(8)=253. Continue until you find that for r = 11, w = 39+D.C., r_{1} + r_{2} +...+ r_{w} = 188.

Thus, the ratio (Total votes in 39 states + D.C.)/(Total votes in nation) is proportional to 188/436. But, in each state, only a simple majority is needed (i.e. 0.5(188)+1 = 95). Thus, our final relationship is that (Total votes won in 39 states + D.C.)/(Total votes cast in nation) is proportional to 95/436 = 022%.

Under our assumptions, a candidate could win only 22% of the people's popular vote, win more than 269 electoral votes, and thus win the election!

**Note:** For a more complete analysis of the mathematics underlying this solution, see George Polya's "The Minimum Fraction of the Popular Vote That Can Elect the President of the United States," *Mathematics Teacher*, March 1961, pp. 130-133.