Math and Logic

Who Won the Race?

How much information do you need to find the finish order of a three-person race? It turns out, not much at all.

Suppose that in a race among Amelia, Boris, and Carlos, we only know one fact — that Boris finished before Carlos. What are the possible orders for the entire race? Keep reading to see some reasoning we can use, or jump ahead to today's challenge for a trickier problem.

We can start by thinking about a single unknown factor in the problem: "Where did Amelia finish?" To see this logic demonstrated, use the arrows below the figure:

Since we know the relationship between Boris and Carlos, Amelia's position in the race completely determined the entire finishing order. There are three possible total finishing orders — one for each of the positions in which Amelia can finish.

What if we add another runner to the race, Dara, and have a more complicated scenario?

Dara finished more places ahead of Amelia than Boris finished before Carlos.

We can start by again thinking about an unknown — "How did Boris and Carlos finish?"

There is only one way these conditions can be satisfied without conflicts, so we have only one solution.

In the problem below, you'll be putting together information like this, but with five runners instead of four. Can you use logic to figure out the order?

Today's Challenge

Four runners compete in a race — Amelia, Boris, Carlos, and Dara.

After some confusion at the finish line, it's unclear what the final finishing order was, but this information is known:

  1. Dara finished before Amelia.
  2. Boris wasn't third.
  3. There were two runners between Amelia and Carlos. \\[-0.7em]

\\[-0.5em] Who won the race?