Measure 45 Minutes: The Burning Ropes Challenge

Two ropes, one lighter, and no way to measure by length. Try the classic puzzle first, then check the timeline and the reasoning behind it.

Two ropes. One lighter. A stopwatch you do not get to use. The burning ropes puzzle has been around for a long time, and it still catches people the first time they see it. Read the setup below, then stop and try it before you scroll on.

The Challenge

You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn from one end to the other. The ropes burn unevenly, so half the rope by length is not half the time. Using only these ropes and the lighter, measure exactly 45 minutes.

Two unlit, unevenly thick fuses resting side by side on a dark wooden workbench, with a blurred pocket watch in the background. — Two unlit ropes, each good for one hour. Neither burns at a steady pace.

The uneven burn is the rule that matters most. It rules out the obvious move of cutting off three quarters of a rope and burning that piece. A thick stretch might take 50 minutes and a thin one ten. Length tells you nothing, so the only thing you can trust is the total time a full rope takes.

The usual version uses minutes. Some sites shrink it to a 60-second rope and a 45-second target. The logic is identical, so pick whichever feels better.

Try It First

Give it a few honest minutes. The puzzle is rated 4 out of 10 on difficulty by one quant-interview site, so it is solvable without a trick you have to be told. If you get stuck, use the hints one at a time.

You are allowed to light a rope at either end. Do you have to light only one?
What happens to the burn time if fire starts at both ends of the same rope?
You need a clean 30-minute checkpoint. Which rope can give you one?
While that checkpoint runs, what should the other rope be doing?
Once the checkpoint ends, how much time is left on the second rope, and how can you shorten it?

Most people get stuck at hint three. They look for a way to mark 45 directly. The better move is to build 30 first and then find the last 15 inside what is left.

The Answer

Here is the standard solution. Light one end of rope 1 and both ends of rope 2 at the same moment. When rope 2 burns out, light the other end of rope 1. When rope 1 burns out, 45 minutes have passed. The Wikipedia entry on the puzzle and a GeeksforGeeks walkthrough give the same sequence.

0 minutes: Light rope 1 at one end. Light rope 2 at both ends.
30 minutes: Rope 2 burns out. Rope 1 has 30 minutes of burn left. Light its other end now.
45 minutes: Rope 1, now burning from both ends, finishes. Done.

A dim workshop bench at night. One fuse has burned down to a charred stub, while the other glows orange and sparks at both ends as it burns, with a brass pocket watch nearby. — The 30-minute mark: the first rope is spent and the second is lit at both ends, with 15 minutes to go.

The arithmetic is short. Rope 2 lit at both ends takes half of 60, so 30 minutes. In that same stretch rope 1 has burned for 30 minutes from a single end, which means exactly 30 minutes of burning remain. Lighting the free end doubles the number of fire fronts, so the remainder takes 15 minutes. Thirty plus fifteen is 45.

Why the Uneven Burn Does Not Matter

This is the part people distrust. If rope 1 burns unevenly, how can you know where its halfway point sits? You do not. You never need to. You only need to know how much burning time is left, and that is fixed by the clock rather than by the geometry.

After 30 minutes, rope 1 has used up 30 of its 60 minutes of burn. Whatever flame is still on it represents the other 30 minutes, even if that flame has crept across a short thick stretch or a long thin one. When you light the far end, the two fire fronts work through that remaining 30 minutes of material together and meet in the middle of it. They finish in half the time because they share the work.

One caveat. The sources back the both-ends reasoning with only a short statement, not a full proof, so treat the explanation above as the intuition rather than a formal argument. The puzzle also depends on its stated assumptions. If a rope burned at different speeds in different directions, the trick would break.

Variations and Where It Shows Up

The puzzle turns up in odd places. Wikipedia notes that it is sometimes posed in job interviews and has been suggested for middle school math classes. A January 2025 Guardian piece collected interview questions for aspiring billionaires, and its rope puzzle is presented as a PayPal interview question: two ropes of one hour each, 45 minutes to measure.

Wikipedia also describes an alternative solution that involves lighting a random point on a rope. It gives only a brief description and no worked example, so it is better left as a thing to explore than a method to rely on. If you want a variation to try, change the constraints. Ask whether you may cut a rope, whether the ropes can differ in burn time, and whether you can light a rope somewhere other than an end. Each change opens different answers.

There is a deeper corner too. The same lighting rules lead to what mathematicians call fusible numbers, the amounts of time you can measure this way. Wikipedia lists a few examples: the smallest above 0 is 1/2, the smallest above 1 is 1 1/8, and the smallest above 2 is 2 1/1024. The sources define the idea but do not explain the theory in detail, so that is a rabbit hole for another day.