Home > math puzzle, Teachings > Rope burning logic puzzle (just slightly more generalised)

## Rope burning logic puzzle (just slightly more generalised)

Many seem to know already the famous rope burning puzzle. When we google it, there will be infinitely many websites discussing this puzzle. However, I try to (kind of) generalise this puzzle a little bit and relate this puzzle to understand the difference between axioms, theorems, lemmas and corollaries that are often used in mathematics articles.

Here is the puzzle: There are two ropes and one lighter. Each rope has special properties.

1. If we light one end of the rope, it will take exactly one hour to completely burn out.
2. The density of the rope is not uniform, which means that burning half the rope would not take half an hour.
3. Those two ropes are not identical, they aren’t the same density nor the same length nor the same width.

The question is how do we measure exactly 45 minutes using those two ropes.

We shall not discuss the answer here, instead I would like to give some logical consequences of those rules given in the properties of the rope. Those three properties are called axioms in mathematics article. They are not to be proven. We have to believed them, we have to accept it.

The first consequence is summarized in the following lemma (which is a small relatively easy consequence of the axioms).

Lemma 1
If we burn both ends of the rope, the rope will take 30 minutes to be completely burned

This lemma is very useful and it is used in solving the puzzle. It seems obvious but we can’t just take it for granted. Anyway, here is the proof.

Proof:
Let us prove this by a contradiction. Assume it would take $x$ minutes by burning both ends of the rope until it is completely burned. Assume $x \neq 30$. Let us consider the case where $x > 30$.  At first ($t=0$) the rope is burned at both ends and after $x$ minutes the rope would be gone. However, after 30 minutes there is still a segment of the rope that hasn’t been burned yet. See the figure below.

And now, let us consider if at first the rope is burned only at one end. It would take then 30 minutes until the segment AB is completely burned and it would take another 30 minutes to completely burn the segment CD. A contradiction, as it would need more than one hour to completely burned out the rope. The case where $x < 30$ can be proven similarly. QED

The next consequence is summarized in the theorem below. Theorem usually used to give a significant consequence of the three axioms given above. The theorem tells us that the solution of the puzzle exists.

Theorem 2
There is such a way to measure 45 minutes using only two ropes

Proof: Burn both ends of the first rope and burn one end of the second rope. According to lemma 1, it will take 30 minutes to completely burn out the first rope. The next step is to burn the other end of the second rope, therefore it will take 15 minutes to burn out the second rope if we apply lemma 1 once more to the second rope which has 30 minutes remaining rope.  QED.

The puzzle stops here, but our discussion does not. I would like to discuss if we have $n$ ropes then how many minutes we can measure accurately. It turns out this problem is an immediate consequence of the last theorem.

Corollary 3
Suppose there are $n$ ropes that satisfy the properties given in the beginning of this section, then there is such a way to measure exactly $(60 - 60/2^n)$ minutes.

Proof:
This can be shown by inductively applying theorem 2 $n$ times to $n$ ropes.

The next results given below are dealing with only one rope. As given in the Lemma 1, we can measure exactly 30 minutes with using only one rope. However, there are various ways how we measure exactly 30 minutes. I won’t present the proofs here as it’s just fun to figure it all ourselves.

Lemma 4
Let us take the rope that has special properties given above. Cut the rope into two pieces and burn both ends of the first piece and then burn both ends of the second piece. It takes exactly 30 minutes until the rope is completely burned.

Lemma 5
Let’s take the rope that has special properties given above. Cut the rope into $n$ pieces. Burn both ends of the first piece and then burn both ends of the second piece and so on until the last piece. It takes exactly 30 minutes until all the pieces of the rope are completely burned.

All the results above are resulted from burning the end of the rope. We are now asking ourselves what do we get if we burn the rope somewhere in the middle. Logically, the fire will spread in two directions. Depending on the density of the rope, one end of the rope will be caught by the fire first.

Lemma 6
Let’s take the rope that has special properties given above. Burn the rope at one point in the middle. Once one end of the rope is caught by the fire, burn the other end of the rope. It takes exactly 30 minutes until the rope is completely burned.

Of all results presented above, with using only one rope, we can measure exactly 30 and 60 minutes. But I would like to ask whether there is any other time that we can measure using only one rope.

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1. August 25, 2012 at 6:32 pm

Lemma 1 is wrong. If the burn rate is not uniform, one cannot tell how long it takes for ihe rope to burn if lit on both sides.

Microsoft, and all other companies using this type of a riddle to base employee hiring decisions, would serve their stockholders better if they used serious established recruiting techniques for interviewing candidates. It is pathetic what is happening in some of the large Silicon Valley companies.

2. October 10, 2012 at 6:44 am

Thanks – it is definitely important to clarify the assumptions here. This page does a good job of that:

http://www.programmerinterview.com/index.php/puzzles/2-fuses-problem-measure-45-minutes/

• October 10, 2012 at 10:54 am

Thanks for the link, and for all the suggestions, I should have realised that burn rate has to be assumed uniform.

3. November 25, 2012 at 1:56 am

Actually you’re wrong MB. Lemma 1 is correct – even if the burn rate is non-uniform, the fact that it burns in 1 hour when lit from 1 end implies that it burns in 30 minute when lit from both ends. The proof is left as an exercise to the reader.

• February 15, 2014 at 8:31 pm

Matthew Wetmore, if you actually try to prove lemma 1, you might see the impossibility, absent a further assumption.

4. March 24, 2015 at 4:46 pm

Lemma 1 is correct, but the proof is overkill.

Here is rope 1:

A…………B

Light end A. After 30 minutes, the flame will be at some point C:

A,,,,,,,,,,C…B

i.e., AC takes 30 minutes to burn. Because the whole rope takes 60 minutes to burn, the remaining length, BC, must take 30 minutes to burn. Therefore if the rope is lit at both A and B, after 30 minutes the flame will meet at C.