## Two Fuses

You have a lighter and two fuses that take exactly one hour to burn, but they don’t burn at a steady rate. For example, one fuse could take 59 minutes to burn the first inch and then burn the rest of the fuse in the last minute.

How would you use these two fuses to measure 45 minutes?

Light the first fuse on both ends and the second fuse at only one end. When the first fuse burns out you know 30 minutes have passed. Light the other end of the second fuse and when it burns out, 45 minutes have passed.

Posted in Brain Teasers

Nick says
September 8, 2014 @ 19:42

I love how you just state, “When the first fuse burns out you know 30 minutes have passed.” As if this follows from the puzzle statement!

It does follow from the puzzle. The fuse burns out in exactly 60 minutes. Lighting it from both ends means it will burn out in exactly 30 minutes.

Nick says
September 9, 2014 @ 19:13

How does it follow? What assumptions are you making? I realise that 60/2 = 30, by the way!

There are no additional assumptions required. Everything you need is stated in the brain teaser. The fuse burns in 60 minutes if lit from one end. If you light it from both ends, it has to finish in exactly 30 minutes.

Nick says
September 10, 2014 @ 11:07

There is an additional assumption. Consider the fuse AB. If you light it at A, after 30 minutes the flame has reached X. Because the fuse burns out in 60 minutes, we know it must take a further 30 minutes to burn from X to B. But does that necessarily mean it must also take 30 minutes to burn in the other direction, from B to X? No! Not without some kind of additional assumption.

No matter how fast or slow the fuse burns at any given point, lighting it from both ends will always take 30 minutes. Remember, you can’t add your own assumptions to what is stated, such as the fuse having a direction. You can only go on what’s stated in the teaser.

The fuse burns in 60 minutes. No matter how fast or slow it burns at any given point, all of those points will have been covered if lit from both ends in 30 minutes.

For example, if the first half of the fuse takes 59 minutes to burn, the other half must take 1 minute to burn. There’s no other possibility. If you aren’t convinced, see if you can come up with a counter example where fuse AB burns in 60 minutes, but lighting it from both ends doesn’t burn in 30 minutes. And you can’t add any of your own assumptions :)

Nick says
September 11, 2014 @ 07:04

Here’s a counter-example…

Consider a fuse AB that burns as follows:
A to X: 29 minutes
X to B: 31 minutes
B to X: 29 minutes
X to A: 31 minutes

If you light the fuse at A or at B it will take one hour to burn. However, if you light both ends of the fuse, it will burn in only 29 minutes.

I’m not adding any assumptions here! I’m simply providing an example of a fuse that behaves as per the puzzle statement; i.e., takes one hour to burn when lit from either end. It’s you who has to make an extra assumption – not stated in the puzzle – in order to prove that it must burn in 30 minutes when lit from both ends!

Perfect. The major assumption you’ve made is that a fuse has a direction. This is not stated in the teaser and contradicts reality, where fuses have no direction. The material they’re made up of burns the same way regardless of the side you happened to start at. In order for your assumption to be be true, we have to depart from reality.

Assumptions that depart from reality, such as gravity not being in force or a magical fuse that knows which side you lit and burns differently as a result, need not be stated. That would make for very boring and tedious teasers.

Unless an exception to how the real world works is clearly stated, reality is in force. I hope that clears things up for you.

Nick says
September 11, 2014 @ 15:37

Speaking of reality, how many fuses “take 59 minutes to burn the first inch and then burn the rest of the fuse in the last minute.”?? On the other hand, it’s quite plausible that a fuse would burn slightly faster in one direction than the other, perhaps because of some asymmetry of construction. You seem to be proposing a fuse which might burn very slowly for almost the whole hour before dramatically speeding up, and assuming that a fuse with such extravagant properties would always burn at the same rate in both directions. You’re the one making assumptions, not me!

I’m not talking about suspending gravity or any such ridiculous notion, I’m talking about something central to the puzzle: the burn rate of the fuses. Your puzzle statement says the fuses “don’t burn at a steady rate.” I’m taking that statement seriously, and assuming nothing about the burn rate other than that the fuses “take exactly one hour to burn.”

Hah, I wondered if you would be convinced :)

Yes, the burn rate is odd, but it’s not impossible, and it certainly doesn’t mean you throw everything else you know about fuses out along with it.

The first inch of the fuse could be composed of a slow-burning material, while the remaining section is composed of paper. Can you think of a way to build a fuse would burn at different speeds when lit from different directions? I can’t.

The assumption you seem to be making is that an odd burn rate implies that it doesn’t have to follow any other standard behavior. I’m saying an odd burn rate is possible. One could make such a fuse. And an odd burn rate doesn’t negate the rest of the observable behavior of a fuse.

I think the core of the disagreement is that you’re saying you can only know the fuse takes one hour to burn, and nothing else. I’m saying you know the fuse takes one hour to burn, but you can also rely on how fuses work.

We may need to agree to disagree on this one.

Out of curiosity, how would you use the two fuses to measure 45 minutes?

Nick says
September 12, 2014 @ 09:37

How would I use the two fuses to measure 45 minutes? I’d use the method you describe – I think it’s likely to measure quite close to 45 minutes for most fuses!

One way to build a fuse that burns faster in one direction than the other might be to have frayed ends sticking out in one direction only. Let’s say the frayed ends are sticking out to the right. When burning from right to left, the flame could catch the frayed ends and thus lead to the overal flame progressing faster than when burning left to right. However, I’ve not checked this experimentally!

Another possibiliity would be if the fuse were placed over a mound, in an inverted V-shape. In that position, if it takes one hour to burn end to end, it would probably burn out faster when lit from both both ends because both flames would be burning upwards.

Your point that, “an odd burn rate doesnâ€™t negate the rest of the observable behavior of a fuse”, is a good one. I just wonder how common it is for there to be small directional differences in burn rate for fuses/ropes/strings. It’s not an easy question to google!

I’m curious to know if the frayed ends method would work, but I don’t have easy access to fuses to fray :) If you ever do the experiment, let me know. I may have to update this!

I agree with you on it being difficult to google. I’ve searched for a definitive answer about how fuses burn, specifically if they burn differently in different directions but came up empty. The electrical fuse and fuse TV only serve to muddy the waters.

Acutally just had this one on a job interview. Small change is they said – after asking – that you had as many lighters as you want. They wanted to know how to determine your 45min point. Answer – light the middle of one fuse, and only 1 end of the other – simultaneously. Once the first fuse which was lit in middle burns out, you will be at 30 minutes. Then you would need to light the other side of the remaining fuse (which is still burning) and in theory it will then burn out in exactly 15 minutes becuase both ends are burning.

The trick to this one is that time and fuse length are not correlated in any matter, which means that the last 25% of a fuse could burnoff in 1 minute or 10. You just dont know.

GraceyCakes says
November 22, 2016 @ 19:50

The answer makes sense, because lighting a fuse from both ends that takes 60 minutes to burn out will cause both ends to shrink, no matter how fast. My point IS that if one end of the fuse, in 30 minutes, burns 75% of the way, and the fuse ALWAYS takes 60 minutes to burn out, then in 30 minutes the other 25% of the fuse will burn out, and (of course) 75+25=100, or 100%. Therefore, there is no more fuse left, and no matter how fast the fuse burned from either end, 30 minutes will have passed. Then, after 30 minutes, you light the other end of the 2nd fuse, because the 30 minutes left can be split into two, producing 15. This shows that in 15 minutes, the second fuse will burn out. Here’s a percentage example: 30% of it burns within 30 minutes when the fuse is lit from one end. This leaves 70% left, which can be split into two, creating two 35%s. Each of these 35%s take 15 minutes to burn. 35+35+30=100, or 100%. OR the 70% can be split into 40% and 30%, but each of those sections of the fuse will take 15 minutes to burn.

Oh crap, I wrote way more than I meant to. Oh well. R.I.P. my brain.