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Broken Stick Triangle: Puzzle and Step-by-Step Solution

  • Probability
  • Hard
  • Reported at Citadel, Two Sigma, Hudson River Trading, Jane Street

The puzzle

A stick of length 1 is broken at two points chosen independently and uniformly at random along its length. What is the probability that the three pieces can form a triangle?

Short answer: 1/4.

Step 1: Turn the triangle into a condition on lengths

Three lengths a, b and c form a triangle exactly when each is less than the sum of the other two. Because a + b + c = 1, the condition a < b + c is the same as a < 1 - a, which is a < 1/2. So the pieces form a triangle exactly when every piece is shorter than half the stick.

That reformulation is the whole trick. Once the condition is 'no piece longer than 1/2', the rest is geometry.

Step 2: Set up the sample space

Let the two break points be X and Y, independent and uniform on [0, 1]. The pair (X, Y) is uniform on the unit square, which has area 1. By symmetry, handle the half where X < Y, which has area 1/2, and double nothing at the end because the other half behaves identically.

With X < Y the pieces are X, Y - X and 1 - Y. The conditions are X < 1/2, Y - X < 1/2, and 1 - Y < 1/2, which is Y > 1/2.

Step 3: Find the area

In the triangle 0 < X < Y < 1, the three conditions cut out a smaller triangle with corners (0, 1/2), (1/2, 1/2) and (1/2, 1). Its legs are 1/2 long, so its area is 1/2 × 1/2 × 1/2 = 1/8.

The probability is the favourable area divided by the area of the half-square: (1/8) / (1/2) = 1/4. The other half, where Y < X, gives the same ratio, so the answer is 1/4.

Step 4: A useful sanity check and a common variant

One quick check: the probability that one specific piece exceeds 1/2 is 1/4 for each of the three pieces, and at most one piece can exceed 1/2, so P(no triangle) = 3 × 1/4 = 3/4. That confirms 1/4.

In the variant where you break the stick once, then break the longer piece at a uniform point, the answer changes to 2 ln 2 - 1, about 0.386. Interviewers sometimes ask it straight after, to see whether you redo the set-up rather than reuse the number.

Why interviewers ask it

The key move is the reformulation to 'every piece under 1/2'. Draw the unit square out loud; interviewers grade the picture as much as the 1/4.

Knowing the proof is half of it. Work Broken Stick Triangle interactively, then take it under a clock with the rest of the Outcry simulators.

Unlock the Broken Stick Triangle simulatorAll brain teasers

Related Firm Questions

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