Let’s take a break from the presidential campaign to consider a recreational math question posed by New York Times correspondent Binyamin Appelbaum. On Twitter, he wondered:
Is there any number of electoral votes between 0 and 538 that is impossible to amass, assuming electors are faithful?
Put another way, could a presidential candidate win any number of electoral votes from 0 to 538?
The answer is intuitive if you know a key piece of electoral college trivia. But there’s still a fun question of how best to actually prove that intuition.
My best attempt below. Stop reading now if you want to figure it on your own.
Each state gets at least three electoral votes. Most states give all those votes to the winner of the state’s popular vote. So you might think that it isn’t possible for a candidate to get 1 vote, 2 votes, 536 votes, or 537 votes.
That would be true except that Maine and Nebraska allocate votes by congressional district (plus two for the state as a whole). Since Maine has 4 votes, it could provide 1 vote for a candidate.* With 6 votes, Nebraska could produce 1 or 2 votes for a candidate.
So 1, 2, 536, and 537 are possible.
What about the rest of the possibilities? Intuitively, it seems as though all the other numbers should be possible. But how best to prove it? Working through every number would be tedious.
Happily, there are ways to trim it down.
First, you don’t have to do all the cases. It suffices to get vote totals from 0 to 269. If those are possible, so are the rest. Why? Because in those cases, the candidate’s opponent gets between 0 and 269 votes. For a candidate to get 338 votes, for example, their opponent (or opponents) must get 200 votes.
That observation cuts the work in half.
Second, we can get any vote total up to 255 using a little binary reasoning. We can get 1 vote from a district in Maine; 2 from two districts in Nebraska; 4 from Rhode Island; 8 from Kentucky; 16 from Michigan; 32 by combining Illinois (20) and Washington (12); 64 by combining California (55) and Mississippi (9); and 128 by combining Texas (38), Florida (29), New York (29), Pennsylvania (20), Nevada (6), and Utah (6).
Those states can then be combined to create any vote total from 0 to 255. For example, 92 = 64 + 16 + 8 + 4 = California, Mississippi, Michigan, Kentucky, and Rhode Island.
Toss New Jersey (14) into the mix, and you can get any vote total up to 269. QED.
Bottom line: Yes, a candidate can win anywhere from 0 to 538 electoral votes.
* Maine could yield 2 votes for a candidate in a three-way race.
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