# How to rephrase this puzzle about a meditating monk and incense

Here's a math-flavored puzzle I found on Andrej Cherkaev's website that I like to give to my undergraduate math students. I've paraphrased it to try to make it clearer:

A monk was instructed by his teacher to meditate for exactly forty-five minutes. But the monk didn't have a watch or a clock to use to time himself. At realizing the monk's problem, his instructor handed the monk two incense sticks that each take exactly one hour to burn. The incense sticks however are not identical, and they each burn at a non-uniform rate, since after all they are hand-made. Using these incense sticks and some matches, how can the monk arrange for exactly forty-five minutes of meditation?

I'm still not happy with the way this is worded though. Some of my students get stuck on superficial details and stuff like these

• (The major issue) Many of my students don't know what I mean by the sticks not burning at a non-uniform rate. I feel like it detracts from the flow of the statement of the puzzle to have to stop and explain what that means (although they should really know that ... ), and honestly the phrase "non-uniform rate" doesn't feel right in the statement anyways.

• Some students don't know what an incense stick is, which is frankly pretty important (hint: this puzzle wouldn't work with candles).

• There are some complaints along the lines of, "how can the monk even notice the incense sticks are done burning if he's busy meditating?"

• Some students focus too much on the matches (hint: they're only there to light the incense). I think that if I don't need to mention the matches though (if a student knows what an incense stick is, they'll know they have to burn it, right?), so this is an easy fix.

Is there another way to phrase this puzzle, like another story or situation to base this same puzzle on, that would avoid the above issues?

• The puzzle has been asked before, so reposting would just result in a duplicate. I'm not sure about the tags, I'll let someone else advise you on that. Welcome to Puzzling! – boboquack Feb 7 '18 at 5:42
• Do not use monk, just give the quiz like this : A king want to test his son. The prince have to measure 45 minutes time using 2 ropes. but the prince can not cut the rope. – Jamal Senjaya Feb 7 '18 at 6:11
• For reference, here's the version of this puzzle on PSE (I don't think that version has particularly great phrasing, but may help with ideas). – Alconja Feb 7 '18 at 6:24
• @JamalSenjaya That removes the whimsical feel from the puzzle though. There's a monk, he's got to meditate, and there's even incense involved. There's a nice theme happening. Having it be about a king and his son doesn't add to any story. I might as well rephrase it as "Suppose you have two pieces of rope, ... How do you measure 45 minutes," which is way less stimulating than a puzzle in a story about a meditating monk. – Mike Pierce Feb 7 '18 at 6:57
• @Chowzen right, so part of the challenge is supposed to be how to overcome that non-uniformity (i.e. you need to treat each stick as an indivisible unit). Not sure how important it is to the asker, since you could put the same kind of restraints on question by saying something like "the monk didn't break the sticks." – Guest Feb 7 '18 at 14:48

I agree that "non-uniform" is a bit too much of a technical term to fit well with the tone of this puzzle. Here's my suggestion; I've also tightened up the phrasing/language in a couple of places, and fiddled with the flavour text in an attempt to make it explain the restrictions on the monk without introducing too much irrelevant information.

A monk needs to meditate for exactly forty-five minutes, but - living in an abbey - he doesn't have a watch or a clock with which to time himself. All he has is two incense sticks, which he knows each take exactly one hour to burn. Unfortunately, being hand-made, the incense sticks aren't identical to each other, and they're imperfectly shaped so that he can't rely on a stick burning at the same rate all the time.

Using these incense sticks and some matches, how can the monk arrange for exactly forty-five minutes of meditation?

• If you're worried about your students not knowing what incense sticks are, you could replace them by e.g. ropes (as in Puzzling SE's version of this puzzle) or wooden rods.
• Don't worry too much about people saying "how can he notice if he's busy meditating?" That kind of nitpick isn't in the spirit of the puzzle, and besides it's easy to notice if, for example, the burning sticks are the only source of light, so that as soon as the flame goes out he's plunged into near-darkness (just enough light for him to be able to light the other stick).
• If students focus too much on the matches, then you've planted a successful red herring. You could just remove the matches from the puzzle as you say, but you could also leave them in to make it slightly harder for people being too clever for their own good.
• If someone asked me this question, I would assume that the monk can't do anything else while he's meditating, like fiddle with incense. I would assume he somehow needs to get it set up at the start, and likely wouldn't ever consider the intended solution. This wouldn't be because I'm being nitpicky, or trying to find faults with the puzzle, it would just be a genuine (and wrong) interpretation of the puzzle I'm supposed to solve. – Guest Feb 7 '18 at 20:12
• I really like the way yours is worded. I still worry that some students won't fully grasp what "... he can't rely on a stick burning at the same rate all the time." means, but I think we can just attribute that to careless reading, or the students not thinking too hard about the puzzle. And your phrasing does fit with the tone much better. – Mike Pierce Feb 7 '18 at 21:33
• And I'm not sure that the matches are necessarily a red herring. One of my student had the thought that a match would probably burn at a constant rate, so they suggested you could burn matches successively while an incense stick is burning to measure how many matches burn per hour, and then burn three-quarters that number of matches to measure 45 minutes. :P – Mike Pierce Feb 7 '18 at 21:35
• @MikePierce Maybe the monk is using a cigarette lighter instead of matches? :-) – Rand al'Thor Feb 8 '18 at 0:03
• So I've field-tested this statement of the puzzle with my students, and they're still not grasping the bit about the sticks not burning at a constant rate. About fifteen students thoughtfully responded to the puzzle, and each of them wanted to either break the sticks or measure out 3/4 of a stick. So I'm considering adding an explicit sentence right before the actual question, "for example, if you were to break a stick directly in half, you can't guarantee that it burns for exactly half and hour." – Mike Pierce Apr 19 '18 at 23:08

The "how can the monk even notice the incense sticks" responses sound like more than nitpicking. Doesn't the solution involve more than just noticing the incense sticks? He actually has to interact with them during that 45 minutes, doesn't he? Presumably he can't do that while he is busy meditating (he can't arrange for 45 minutes of meditation, because his meditation would be broken by getting up to mess with incense sticks).

Unless the solution you're looking for involves laying the two sticks on the floor in a T shape and then lighting all 3 ends. If that's the case, you might need to be specific enough about how the sticks burn that the students will know this is possible (they need to be sure that one stick will light another, and that they will burn properly laying down).

Of course that wouldn't work if they burn at a non-uniform rate, but you can have them burn uniformly (avoiding questions about what "non-uniform" means) and instead stipulate that the monk is not allowed to break the sticks in two. He's a monk, he'll be used to having arbitrary restrictions placed on him. And he'll be meditating (with his eyes closed) so he can't just burn one stick and watch it until only a quarter of it is left. You can read this all into the story...

Without breaking the sticks, the monk lays the incense on the floor, lights it, and closes his eyes to meditate. 45 minutes later, he no longer smells the incense burning, and knows his meditation is complete. How did he manage this?

You may also want to say that the incense sticks are entirely coated with the flammable incense. The ones I've seen always have a plain wood stick at one end, and it doesn't burn as easily as the incense end (can easily go out). Otherwise, I doubt anyone is likely to think of lighting it from that end.

Maybe you can also give some other clues. You say the sticks take "exactly one hour to burn," but what does that mean? If I throw them in the fireplace, I bet they'll burn faster than that. Maybe instead you can say something a little more specific, like "once one end of the incense stick is lit, it will take exactly one hour to burn down to the other end." That might get them going in the right direction.

• Love that proposition, The non uniform idea is complex to understand, quite unnatural too and here we don't need it. Saying he doesn't smell incense after 45 min ask the question of how is that possible and should give the idea of lighting from two sides. It is not exactly the same problem though! – Untitpoi Feb 7 '18 at 14:03
• All the incense sticks that I have seen only light from one end, the other end is firstly not covered in incense and secondly used to support the stick while it burns - it is not practical to light both ends – Lee Leon Feb 7 '18 at 14:51
• @LeeLeon, agree 100%. Might be a regional thing, but they're all just wooden sticks at one end here. Conceivably the whole thing could be coated with the scented stuff, but asker should definitely specify that. – Guest Feb 7 '18 at 14:54
• @LeeLeon, yeah, the traditional solution would require him to get up after 30 minutes; that's why I'm suggesting the T configuration, uniform burn speed, and explaining everything else away in the story (for example asker can just say the meditation heightened the monk's awareness to the point where he could immediately detect that there was no more smoke, or whatever). – Guest Feb 7 '18 at 15:37
• @MikePierce, right, the top end of the vertical stick of would have to be smack in the middle of the horizontal stick for it to work. Of course that's assuming you scrap the non-uniform burning. I figured eyeballing it would be good enough for government work, but I guess you could give him a tape measure, or a piece of string he can fold in half, or something.  If students still struggle with this, I'd suggest "split-testing" it against formulations where it's clear that the things can be lit from both ends without going out, and it's clear the monk can mess with them while meditating. – Guest Feb 8 '18 at 17:10

Since (what I'm assuming is) the intended solution involves lighting the second stick partway through the 45 minutes, it might be better to rephrase the question as:

A monk in training was asked to help his mentor in his meditation. The mentor wants to meditate for exactly 45 minutes, but all the monk has to measure time are two incense sticks, each of which take exactly one hour to burn from end to end. Even worse, the rate at which they burn is irregular! The monk can't tell anything about the percentage of time passed from the percentage of the incense stick burned, other than that the stick will be completely burned through in an hour.

What strategy can the monk use to ensure that he lets his mentor meditate for exactly 45 minutes?

This solves some of the nit-picky story issues with the original phrasing, in that now he can devote his entire energy to watching the incense sticks, and can know when to light the second stick and when the time is completely up.

Ah, one of a nice and classical quiz of all time. Well, if this one can help you, in Korea, this quiz is known as "Fuse Problem", and it's one of the famous quiz for cooperate interview questions. It goes like this;

You have two fuse with exactly same length. (Or, you can just say 'identical') Each fuse takes exactly one hour to burn out completely. But, burning speed of fuse is not static. Sometimes half of fuse can burn in just 10 minutes, sometimes can takes half hour to burn just few centimeters. However, when fuse is burnt out completely, it's exactly one hour. Then, how can you measure 45 minutes using two fuses?

• I don't think "exactly same length" is necessary. – Weijun Zhou Feb 7 '18 at 6:10
• A variation on this theme I seem to recall has 3 fuses, some time to prepare, and you need to create a 15 minute fuse. 3 fuses and 1:45 to detonation was another. (One less intuitive variation was burning ropes.) – Bass Feb 7 '18 at 6:16
• At least by how I think of what a fuse is, this would be more confusing than what I point out in the second bullet-point above. I imagine a fuse "leading into something," like a candle, a bomb, a firework, etc. I would never have thought it was alright to light a fuse at both ends. – Mike Pierce Feb 7 '18 at 7:02
• @MikePierce Hmm. you're right, but this version is what I remember. Plus, I think it's okay to light both ends of fuse if there are "only fuse", without any explosives. :) – Kamome Feb 7 '18 at 7:11