Broken Stick Puzzle Answer: 1 in 4 [Solved]

The broken stick puzzle answer is 1 in 4. Cut a stick at two random points and the three pieces form a triangle only a quarter of the time. In the other three cases out of four, one piece is too long, and no triangle is possible.

Here is the question, as it appears in quant and machine-learning interview collections. Take a stick of length 1. Choose two cut points independently and uniformly at random along it, then cut. What is the probability that the three pieces can form a triangle?

Two words in that question carry the whole puzzle: independently and uniformly. Each cut point can land anywhere on the stick, and neither cut knows where the other one landed. Pin that procedure down first, because a different breaking procedure gives a different answer, and that is exactly the trap this question sets.

The Broken Stick Puzzle Answer in One Picture

Three lengths can form a triangle only when each one is shorter than the sum of the other two. That is the triangle inequality, and it is the only rule this puzzle needs. The three pieces add up to 1, so each piece must be shorter than 1 minus itself. In plain terms, no piece may be longer than 1/2. If one piece exceeds half the stick, it is longer than the other two pieces put together, and they can never close around it.

So the question becomes a simpler one. When you pick two random cut points, how often is every piece shorter than half the stick? Draw a picture to answer it. Put the first cut point on the horizontal axis and the second cut point on the vertical axis. Every pair of cuts is one point in a square, and every point in the square is equally likely.

Now mark the good region. Suppose the first cut lands in the left half and the second in the right half. The left piece is under 1/2, and the right piece is under 1/2, so the only danger is the middle piece. The middle piece stays under 1/2 when the two cuts sit within 1/2 of each other. That condition carves out a small triangle inside the square, with corners at (0, 1/2), (1/2, 1/2), and (1/2, 1). Its area is 1/8 of the square.

The mirror image gives a second small triangle for the case where the first cut lands in the right half and the second in the left half. Its area is also 1/8. Add the two good regions together and you get 1/4 of the square. Since every point in the square is equally likely, the probability is 1/4. The broken stick puzzle answer is 1 in 4, and the proof is a picture, not a page of integrals.

The Broken Stick Puzzle Answer, Worked With Numbers

A picture is convincing, but numbers make it stick. Take a stick of length 1 and cut it at 0.3 and 0.7. The pieces are 0.3, 0.4, and 0.3. Each piece is under 1/2, so they form a triangle. Check it against the rule: 0.3 plus 0.3 is 0.6, which beats 0.4, and 0.3 plus 0.4 is 0.7, which beats 0.3. The three sides close up.

Now cut the same stick at 0.1 and 0.8. The pieces are 0.1, 0.7, and 0.2. A middle piece of 0.7 is longer than half the stick. The other two pieces add up to 0.3, so they cannot span it. No triangle is possible, and no rearrangement of the pieces changes that.

There is also a shortcut that skips the picture entirely. Ask where the two cuts fall relative to the midpoint of the stick. Half the time, both cuts land on the same side of the midpoint. When that happens, one piece is guaranteed to exceed 1/2, so the triangle is already dead. That accounts for half of all outcomes.

The other half of the time, the cuts land on opposite sides of the midpoint. Then the two end pieces are both under 1/2, and only the middle piece can spoil it. By symmetry, the middle piece exceeds 1/2 in exactly half of those cases. So the failure rate is 1/2 plus half of 1/2, which is 3/4. The success rate is the rest: 1/4. Two routes, one answer.

Where the cuts fall	Chance	Triangle possible?
Both cuts on the same side of the midpoint	1/2	Never: one piece exceeds 1/2
Cuts on opposite sides, middle piece over 1/2	1/4	Never: middle piece too long
Cuts on opposite sides, middle piece under 1/2	1/4	Yes

The Trap: Say the Procedure Before You Compute

The most common wrong answer to this puzzle is not a wrong number. It is a right number for a different procedure. The 1 in 4 answer assumes two cuts chosen independently at random on the whole stick. Change the procedure and the answer changes with it.

Suppose instead you break the stick once, then break the longer of the two resulting pieces at a random point along it. That sounds like the same puzzle, but it is not. The first break already biases the second one, because the longer piece is where the second cut must land. Worked out exactly, the probability of a triangle under this procedure is 2 times the natural log of 2, minus 1, which is about 0.386, or close to 39 percent. A computer simulation of two million trials lands at 0.3862, matching the formula.

A strong candidate names this gap out loud before doing any arithmetic. Say which procedure you are solving, and ask the interviewer to confirm it if the question is vague. Interviewers set this trap on purpose. They want to see whether you define the random process before you compute with it, because in real work the model is the hard part and the arithmetic is the easy part.

The same warning applies to the simulation check. If you test your answer with code, simulate the exact procedure the question describes. Two independent uniform cuts give 0.25. The break-the-longer-piece procedure gives 0.386. A simulation that answers a different question than the one you were asked proves nothing.

What the Interviewer Is Testing in the Broken Stick Puzzle

This puzzle rewards four skills at once.

1. Whether you turn a shape problem into a number problem. The triangle inequality reduces the whole puzzle to one condition: no piece longer than 1/2. The candidate who finds that condition first has already won half the battle.

2. Geometric probability. Two random cut points are one random point in a square, and probability becomes area. Candidates who can draw the sample space solve in minutes what algebra alone makes painful.

3. Precision about the random procedure. The 1 in 4 answer and the 0.386 answer are both correct, for different procedures. The interviewer watches whether you notice the ambiguity and pin it down instead of charging ahead.

4. Checking your own work. A quick simulation or a worked example with real numbers catches a wrong answer before the interviewer does. The candidate who says “let me test this with 0.3 and 0.7” looks careful, not slow.

Try Another Puzzle Next

If this one made you pause, try the two envelopes paradox answer next: two envelopes, one worth twice the other, and an argument for switching that looks airtight until you check what one letter means. Then test your probability instincts against the birthday paradox answer, where just 23 people in a room make a shared birthday more likely than not.

Is the broken stick puzzle answer really 1 in 4?

Yes, under the standard version of the question: two cut points chosen independently and uniformly at random on the stick. The good outcomes cover exactly 1/4 of the unit square of possible cut pairs. A simulation of two million trials gives 0.2498, matching the exact answer.

What if I break the stick one piece at a time?

Then the answer changes, because the procedure changes. Break once, then break the longer piece at a random point along it, and the probability rises to about 0.386. Always state the procedure first, and confirm it with the interviewer if the question leaves it open.

What happens with four pieces instead of three?

Break a stick into four pieces at three random points, and the chance they form a four-sided polygon is 1/2. In general, n random pieces form an n-sided polygon with probability 1 minus n over 2 to the power of n minus 1. For three pieces that gives 1 minus 3/4, which is 1/4, the same answer as above.

Source: the question appears in quant and machine-learning interview collections, including the ML Interview Q series on geometric probability. The solution and all text here are original.