**Question:**

There are three ants at the corners of a regular triangle. Each ant starts moving on a straight line toward another, on a randomly chosen course. What is the probability of avoiding an ant pileup?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

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**Solution:** Here are three different responses from three candidates. Note that while all come to the same answer, their approaches are totally different. Most interviewers agree that the approaches are more important than the answer. Candidate A’s response is clearly explained. Candidate B responds with a story, even assigning a name to the ant. Candidate C is clearly the most rigorously educated. Which candidate would you prefer on your team? The important thing is to think out loud so the interviewer can follow your reasoning.

Candidate A: Each ant can move only in two directions. Multiply the number of ants by the number of directions (2 2 2 = 8 ways) to get the total number of possibilities. Now, there are only two ways the ants can avoid running into each other. Either they all travel clockwise, or they all travel counterclockwise. Otherwise, there has to be a collision. Half the possibilities will result in collisions; half will result in no collisions. So the answer is half of 8, or 1 in 4, or 25 percent.

Candidate B: Select one ant and call him “Willie.” Once Willie decides which way to go (clockwise or counterclockwise), the other ants have to go in the same direction to avoid a collision. Since the ants choose randomly, there is a 1-in-2 chance the sec-ond ant will move in the same direction as Willie, and a 1-in-2 chance the third ant will do the same. That means there is a 1-in-4 chance of avoiding a collision.

Candidate C: Consider the triangle *ABC*. Let’s assume that the ants move toward different corners along the edges of the triangle. The total number of movements is eight (*A*→*B, B*→*C, C*→*A A*→*B,* *B*→*A, C*→*A A*→*B, B*→*A, C*→*B A*→*B, B*→*C, C*→*B A*→*C, B*→*C, C*→*A A*→*C, B*→*A, C*→*A A*→*C, B*→*A, C*→*B A*→*C, B*→*C, C*→*B*). The number of non colliding movements is two: (2 *A*→*B, B*→*C,* *C*→*A A*→*C, B*→*A, C*→*B*). All ants move in either the clockwise or anti-clockwise direction at the same time. *P* (not colliding) = 2/8 = 0.25.

*Answer: *One-in-four chance or 25 percent of avoiding a collision.