Today it’s one of my all time favourite puzzles…..

There are two lengths of rope. Each one can burn in exactly one hour. They are not the same length or width as each other. They also are not of uniform width (they might, for example, be wider in middle than on the end), thus burning half of the rope is not necessarily 1/2 hour. By burning the ropes, how do you measure exactly 45 minutes worth of time?

As ever, please free to say if you think you have solved it and how long it took, but please no solutions. Answer on Monday!

Oh, and if you are struggling to find something to do on this fine day, here is my science stunts video again!

I know the answer to this one, but only because I had to solve it before during a job interview a few years ago (at a place I still happily work at). I think originally it took me maybe 4-5 minutes to solve in the back of my mind while we talked about other things.

On a side note I’d be interested to know how these ropes were manufactured, as they are non-uniform they could could not simply have been cut to length.

It’s a fun puzzle (I took ~2mins), but Will’s right. How could you manufacture such lengths of rope??

Also, what are the flaws that people are talking about. You have to make some reasonable assumptions to solve the puzzle… but so far I can’t see any flaws.

Took me half an hour to figure it out, though I am not so sure about the answer.

I kinda knew the catch, but hadn’t specifically come across this puzzle. Took me about a minute (mabye 2) to get there though. Quite satisfying.

Interesting that this is on a job interview. Never had an interview where I had that kind of thing. I’ve done “team” exercises, roleplaying and was asked what my favourite movie was, but never a puzzle.

The only times I’ve gotten “puzzle” questions (or even “if you were a tree” style questions) in an interview is when the company had no clue about how the actual job they were hiring for was done. No serious company wastes time presenting toy problems when they could pick your brain about how you’d handle their current operations.

Instant.
a variation of the mosquito coil problem.

I got a solution within a minute or two of seriously thinking about it (or several hours after reading it.) However, although my brain is up to the task, there may be other parts of my body that I don’t possess in sufficient numbers for true, down-to-the-second accuracy.

This is a good example of a logic puzzle that has a neat, satisfying answer, which on reflection turns out to be seriously flawed. Flawed in the sense that there are ropes which burn end to end in one hour, but which would *not* measure 45 minutes if you followed the standard, neat solution. Instead, they might measure 30 minutes or they might measure an hour — oops! We need an extra condition to rule out these pathological ropes, which rather spoils the simplicity of the puzzle statement.

So my counter-puzzle, which I think would make a good interview question, is to construct an explicit counter-example. Once you’ve solved the original puzzle, identify the hidden assumption that your solution relies on, and thereby specify (in terms of their burn rate) ropes which would measure one hour using your solution.

I think I saw the solution straight away to this one – it seems obvious to me, but the fact that it seems so obvious makes me wonder if I have got it wrong after all. I guess I’ll have to wait till Monday to find out.

Allright I got a solution, I think it’s consistent with what Google found for me.
They say everybody in a Wall Street firm knows it 😉

Eh .. clock or watch? I always tend to use that technique when I need to measure time. I know .. it’s very modern and not so common yet but hey .. Garden sun clocks do have their backdrafts in my humble opinion ..

I spotted the common themes with requested time to be measured and the ropes, but I failed to find a way to implement it. After a quick google for the answer I kicked myself for getting so close and not spotting it!

It’s intriguing to see so many people coming up with an answer when actually there is insufficient information. Why do people overlook the flaw in this puzzle? I find that more interesting than the puzzle itself.

I have a solution that I’m satisfied with. It took me slightly less than five minutes, I think. I got a clever answer early on that turned out, on reflection, to be rubbish.

I look forward to seeing this hidden flaw that Nick mentioned! After carefully considering my answer, I’m pretty sure that the only unstated assumption it relies on is one that is very strongly implied by the wording of the problem (and by the absence of an explicit caveat ruling it out).

I never understood this one. Nick is right, the common answer violates the original premise. For those with answers re-read the question with your solution in mind and you should see the fault.

Then this gets interesting when you think about what makes people solve it the way they do and why was it worded the way to guide to a certain solution that is expected. It is a logic puzzle that you need logic to solve but also fails to employ it’s own logic into the solution.

I have no idea what Nick & Chris are talking about with the implicit flaw? Perhaps they’re thinking of a different solution that has a flawed underlying assumption, not sure, but agree with Tim that it will be good to find out what the hidden flaw they’re thinking about might be. One assumption I’ve made is that I have a match to light the ropes, but guess that couldn’t be the flaw, though granted it’s not stated in the question.

I think I see what the supposed flaw might be but, on further consideration, I don’t think it is a flaw. It will be interesting to see the comments after the solution is published.

I’ve got one potential ‘flaw’ in the wording. Not all burning ropes can be set alight instantly – you sometimes need to put a flame to it for some time before it starts burning.

My solution requires that one needs to be able to set the ropes alight immediately (and / or at the same time.)

The hidden flaw I have in mind has nothing do with not having enough matches, or not being able to set ropes alight instantly! It concerns the burn rate of the ropes, so it is directly relevant to the solution and to this puzzle’s status as a “logic puzzle”.

We simply aren’t given enough information about the burn rate to deduce a solution. We can make a simplifying assumption, and thereby deduce a solution. But if you arrive at an answer without noticing the need for the assumption then I don’t think you’ve really solved the puzzle; you’ve just been lucky. Just like in a maths exam, to get full marks you need more than the correct answer, you must also show all your working!

To help identify the hidden assumption, break your solution down into the smallest steps you can and try to justify each step using information from the puzzle statement. You should notice that one of the steps cannot be justified…

Finally, I don’t find the simplifying assumption either reasonable or physically realistic, which, for me, irredeemably spoils the puzzle.

I agree with “Impossibly Stupid” that this would make a poor interview question. Apart from anything else, it’s too well known — just about every tech interview site on the web mentions it.

@Bastiaan, I know this was a Christmas Day puzzle, but I didn’t realise Christ Himself had commented! 😉

I didn’t say that *this* puzzle in particular would make for a poor interview, but that *all* toy problems do. I don’t just mean ones that are a false idealization of the real world, either. I once worked for a blind person who had a bunch of *awesome* wrought iron puzzles that really tested your ability to think spatially and manipulate them with your hands, but I didn’t find out about them until long after we interviewed and discussed the duties of the actual job.

And to your idea that “burn rate” matter, I’m sure the solution everyone else has come up with has the burn rate . . . self regulated. That is the nice trick about this puzzle. Not only do you have to figure out how to regulate the rate, you also have to figure out how to associate two ostensibly unrelated values. While we may all be missing something, consider the possibility that it is you who hasn’t come up with an appropriate solution.

Consider the possibility that division might not be an appropriate operation.

Why do you assume anyone is doing some sort of division? If that’s how *you* came to a flawed answer, I again forward the distinct possibility that others have solved it differently. Your vague assertions certainly aren’t convincing anyone, so we’ll all wait until Monday when you can either say “told you so!” or end up feeling pretty silly for being so smug.

I have to be vague because I don’t want to give away the supposed solution, or the flaw therein. I thought you would have realised that.

As for division, are you sure you’re not using it somewhere in your solution?

I have a nice, logical solution that doesn’t require division and compensates for rope thickness and burn rates.

Peter, maybe you’ve found a different approach, but the standard solution *does* use division.

Hmm, I’ll be interested to see what these flaws are because I can’t see them. In the “common solution” (which I believe is the same as the one I’ve thought of) the burn rate of the ropes simply doesn’t matter. It’s eliminated as a consideration due to the manner in which the ropes are ignited (I hope I’m not giving too much away).

Anyway, it took me a couple minutes to get the “Aha!” and then another minute or two to confirm my solution.

But maybe I’m wrong. I’m looking forward to hearing what the flaws and wrong assumptions are on Monday.

A self lighting match? With a safety match wouldn’t one need a rough surface as well? 🙂

I’m sort of hoping I’m wrong now. I’ve been all over this puzzle and can’t see a problem with my solution. It would be interesting to know what I’ve missed.
Nick’s last comment about division however makes me think that he doesn’t have the same solution as me.

I will follow the lead of several others, stick my neck out, and state my belief that the puzzle as it stands has no solution. Looking forward to being proven wrong!

Having conceded defeat I went searching online for the solution, I take it back! I obtained the bulk of the solution but didn’t apply it, now I’m happy with the solution though. It is a very good puzzle.

I’ve not seen this before and after a minute I concluded that it was not possible. About five minutes later I figured it out. Nice puzzle!

I have the solution that I believe is correct.
To me the real riddle is trying to find out the “flaw” in my answer…
I am doing some division, as Nick stated, but it is VERY simple division.
Simple along the lines of… One hour equals two half hours equals four fifteen minutes… Is this the division you’re referring to Nick?

There is AN assumption I can see, but in the context of a logic puzzle it is a reasonable one, and one that would be very straightforward to explicitely state in the question (and hence can hardly be said to “spoil the simplicity of the puzzle statement”). In saying that the rope burns end to end in an hour, it’s necessary to assume that it doesn’t matter which end you start from.

Here’s an example where this assumption is violated. Suppose that, in minute N+1, either the same volume of rope is burned as was burned in minute N, or else the same length of rope is burned as was burned in minute N, whichever of those two options leads to the greater amount of rope being burned. Now, if you consider a rope that’s thick on one end and thin on the other, then it will burn out sooner if you light it on the thicker end.

But honestly, this is a logic puzzle, not a physics experiment, and the assumption can be stated in just five words: It; Doesn’t; Matter; Which; End. I see no simplicity being spoiled here.

D’arcy, yes, that’s the division I’m referring to. I’m not suggesting you’re making an arithmetical error! I’m asking whether division is necessarily the appropriate mathematical operation to use in this case.

Flesh-eating Dragon, that’s not quite the assumption I’m thinking of. I agree it’s reasonable to assume, even with the existing wording, that it doesn’t matter which end of the rope you ignite. And, as you say, a simple clarification would not spoil the puzzle.

The assumption I have in mind would require a longer, more cumbersome clarification that would unfortunately also give a hint how to solve the puzzle.

Stop . . . wait for it . . . twisting yourselves into knots over this rope puzzle. No assumptions are violated in the volume/length/other department. The width not mattering is stated simply to indicate an unpredictable burn rate; Richard could just as well have stated the ropes were damp in places and would thus burn unevenly. All that matters is that the *total* length will burn in one hour; there is no “burn out sooner”. If you want to take issue with something, wait until Monday when everyone can discuss their full solutions.

The central premise of this puzzle is indeed the unpredictable, uneven burn rate. And that’s what seems to be catching you out. It simply isn’t true that, “All that matters is that the *total* length will burn in one hour…” There’s more to it than that.

While reading the puzzle (which I might have solved before but don’t consiously remember) I started pouring water between pitchers in my mind, and when I came to the “45 minutes” part I had a solution to match it already.

Reading through the comments I try to find this mentioned catch in my reasoning. But searching for hidden assumptions I can only come up with the one Flesh-eating Dragon mentions; that it would burn for one hour from end A to end B or equally from end B to end A. (Otherwise both ropes would have to be marked “Ignite here” or I wouldn’t know if it measured an hour or not.)
Well, I also assume a lot of stuff are irrelevant, like time to ignite or orientation of the ropes, etcetera.

I like how these comments make me doubt and scrutinize the solution I was prepared to take poison on… 🙂

There is also a way for measuring 40 minutes with these ropes. Though you need to make some more assumptions as you do here, but it still stays a logical puzzle.

Took me quite a while. I’d never heard this one before. I came up with three hypotheses that I was able to reject, and then the solution hit me, and it was so obvious. I think it took about 8-9 minutes.

I know the answer to this one, but only because I had to solve it before during a job interview a few years ago (at a place I still happily work at). I think originally it took me maybe 4-5 minutes to solve in the back of my mind while we talked about other things.

I’d have the solution if they were identical

I think I have the solution.

Finally I get one! 1-2 minutes

I’ve seen this before too. I didn’t get it when I first saw it, but I have remembered the solution in about 1 minute. Merry Christmas!

On a side note I’d be interested to know how these ropes were manufactured, as they are non-uniform they could could not simply have been cut to length.

It’s a fun puzzle (I took ~2mins), but Will’s right. How could you manufacture such lengths of rope??

Also, what are the flaws that people are talking about. You have to make some reasonable assumptions to solve the puzzle… but so far I can’t see any flaws.

Took me half an hour to figure it out, though I am not so sure about the answer.

I found a solution within 1Minute.

Merry X-Mas btw 😉

Solved – 30 seconds. Love these puzzles.

MERRY CHRISTMAS!!

A friend told me this puzzle ( he had to solved it for a job interview as well ) and I know it took me about ten minutes to figure it out.

I vaguely remembered this puzzle and knew the catch of it. Still, it took me about 30 seconds to tie the loose ends together and get to the solution.

Finally! A friday puzzle I already know 🙂

It took me about 20 seconds when I first solved it…the one I heard was with candles…

A good one! Never seen this one before (I think…), took maybe 15-20mins of more or less active pondering to reach a solution.

This is tricky, I don’t think I will get it 🙂

I kinda knew the catch, but hadn’t specifically come across this puzzle. Took me about a minute (mabye 2) to get there though. Quite satisfying.

Interesting that this is on a job interview. Never had an interview where I had that kind of thing. I’ve done “team” exercises, roleplaying and was asked what my favourite movie was, but never a puzzle.

Beats Myers-Briggs tests, I suppose.

The only times I’ve gotten “puzzle” questions (or even “if you were a tree” style questions) in an interview is when the company had no clue about how the actual job they were hiring for was done. No serious company wastes time presenting toy problems when they could pick your brain about how you’d handle their current operations.

Instant.

a variation of the mosquito coil problem.

I got a solution within a minute or two of seriously thinking about it (or several hours after reading it.) However, although my brain is up to the task, there may be other parts of my body that I don’t possess in sufficient numbers for true, down-to-the-second accuracy.

On second thoughts … yes, I do.

Wow, I had to let this one hang in the back of my mind for a bit until I figured it out, 100% sure I’ve got it right now! pretty neat 🙂

me too 🙂

at first I couldn’t see the answer..

but I had a feeling that it’s somewhere in the air 🙂

then, I gave it another try and, bum! nice.

I got the technique first thing, but had to come back after a few minutes of being distracted to get the whole process. Clever.

This is a good example of a logic puzzle that has a neat, satisfying answer, which on reflection turns out to be seriously flawed. Flawed in the sense that there are ropes which burn end to end in one hour, but which would *not* measure 45 minutes if you followed the standard, neat solution. Instead, they might measure 30 minutes or they might measure an hour — oops! We need an extra condition to rule out these pathological ropes, which rather spoils the simplicity of the puzzle statement.

So my counter-puzzle, which I think would make a good interview question, is to construct an explicit counter-example. Once you’ve solved the original puzzle, identify the hidden assumption that your solution relies on, and thereby specify (in terms of their burn rate) ropes which would measure one hour using your solution.

I think I saw the solution straight away to this one – it seems obvious to me, but the fact that it seems so obvious makes me wonder if I have got it wrong after all. I guess I’ll have to wait till Monday to find out.

Pretty well known puzzle. Didn’t take me longer than the blink of an eye 😉

Got it, about 30 seconds I think.

I can’t think of a solution. I hope it’s the food and booze– I mean the holiday spirit.

Allright I got a solution, I think it’s consistent with what Google found for me.

They say everybody in a Wall Street firm knows it 😉

Eh .. clock or watch? I always tend to use that technique when I need to measure time. I know .. it’s very modern and not so common yet but hey .. Garden sun clocks do have their backdrafts in my humble opinion ..

I spotted the common themes with requested time to be measured and the ropes, but I failed to find a way to implement it. After a quick google for the answer I kicked myself for getting so close and not spotting it!

Got it, in about 3 minutes.

Score, took about a min. Told it to my family and they got it in a min too

It’s intriguing to see so many people coming up with an answer when actually there is insufficient information. Why do people overlook the flaw in this puzzle? I find that more interesting than the puzzle itself.

Got it in about 45 seconds

I have a solution that I’m satisfied with. It took me slightly less than five minutes, I think. I got a clever answer early on that turned out, on reflection, to be rubbish.

I look forward to seeing this hidden flaw that Nick mentioned! After carefully considering my answer, I’m pretty sure that the only unstated assumption it relies on is one that is very strongly implied by the wording of the problem (and by the absence of an explicit caveat ruling it out).

About a minute. That’s a very clever pussle. I like it a lot.

I never understood this one. Nick is right, the common answer violates the original premise. For those with answers re-read the question with your solution in mind and you should see the fault.

Then this gets interesting when you think about what makes people solve it the way they do and why was it worded the way to guide to a certain solution that is expected. It is a logic puzzle that you need logic to solve but also fails to employ it’s own logic into the solution.

I have no idea what Nick & Chris are talking about with the implicit flaw? Perhaps they’re thinking of a different solution that has a flawed underlying assumption, not sure, but agree with Tim that it will be good to find out what the hidden flaw they’re thinking about might be. One assumption I’ve made is that I have a match to light the ropes, but guess that couldn’t be the flaw, though granted it’s not stated in the question.

Took a few minutes to solve. I don’t see any logical problems with my answer even after re-reading the question.

My guess would be that Nick & Chris do not have the correct solution.

I think I see what the supposed flaw might be but, on further consideration, I don’t think it is a flaw. It will be interesting to see the comments after the solution is published.

My solution requires just one rope, so it must be flawed. (sigh)

I got it fairly quickly….and without Google I’m positive I got the correct (intended) answer….

Having said that, I am very curious to what Christ and Nick think is a flaw in this puzzle….

I’ve got one potential ‘flaw’ in the wording. Not all burning ropes can be set alight instantly – you sometimes need to put a flame to it for some time before it starts burning.

My solution requires that one needs to be able to set the ropes alight immediately (and / or at the same time.)

The hidden flaw I have in mind has nothing do with not having enough matches, or not being able to set ropes alight instantly! It concerns the burn rate of the ropes, so it is directly relevant to the solution and to this puzzle’s status as a “logic puzzle”.

We simply aren’t given enough information about the burn rate to deduce a solution. We can make a simplifying assumption, and thereby deduce a solution. But if you arrive at an answer without noticing the need for the assumption then I don’t think you’ve really solved the puzzle; you’ve just been lucky. Just like in a maths exam, to get full marks you need more than the correct answer, you must also show all your working!

To help identify the hidden assumption, break your solution down into the smallest steps you can and try to justify each step using information from the puzzle statement. You should notice that one of the steps cannot be justified…

Finally, I don’t find the simplifying assumption either reasonable or physically realistic, which, for me, irredeemably spoils the puzzle.

I agree with “Impossibly Stupid” that this would make a poor interview question. Apart from anything else, it’s too well known — just about every tech interview site on the web mentions it.

@Bastiaan, I know this was a Christmas Day puzzle, but I didn’t realise Christ Himself had commented! 😉

I didn’t say that *this* puzzle in particular would make for a poor interview, but that *all* toy problems do. I don’t just mean ones that are a false idealization of the real world, either. I once worked for a blind person who had a bunch of *awesome* wrought iron puzzles that really tested your ability to think spatially and manipulate them with your hands, but I didn’t find out about them until long after we interviewed and discussed the duties of the actual job.

And to your idea that “burn rate” matter, I’m sure the solution everyone else has come up with has the burn rate . . . self regulated. That is the nice trick about this puzzle. Not only do you have to figure out how to regulate the rate, you also have to figure out how to associate two ostensibly unrelated values. While we may all be missing something, consider the possibility that it is you who hasn’t come up with an appropriate solution.

Consider the possibility that division might not be an appropriate operation.

Why do you assume anyone is doing some sort of division? If that’s how *you* came to a flawed answer, I again forward the distinct possibility that others have solved it differently. Your vague assertions certainly aren’t convincing anyone, so we’ll all wait until Monday when you can either say “told you so!” or end up feeling pretty silly for being so smug.

I have to be vague because I don’t want to give away the supposed solution, or the flaw therein. I thought you would have realised that.

As for division, are you sure you’re not using it somewhere in your solution?

I have a nice, logical solution that doesn’t require division and compensates for rope thickness and burn rates.

Peter, maybe you’ve found a different approach, but the standard solution *does* use division.

Hmm, I’ll be interested to see what these flaws are because I can’t see them. In the “common solution” (which I believe is the same as the one I’ve thought of) the burn rate of the ropes simply doesn’t matter. It’s eliminated as a consideration due to the manner in which the ropes are ignited (I hope I’m not giving too much away).

Anyway, it took me a couple minutes to get the “Aha!” and then another minute or two to confirm my solution.

But maybe I’m wrong. I’m looking forward to hearing what the flaws and wrong assumptions are on Monday.

Okay, I didn’t knew this one. Solve it in one minute, eyes in the ceilings. Beautiful Puzzle… really beautiful. I’ll need matches, of course 🙂

A self lighting match? With a safety match wouldn’t one need a rough surface as well? 🙂

I’m sort of hoping I’m wrong now. I’ve been all over this puzzle and can’t see a problem with my solution. It would be interesting to know what I’ve missed.

Nick’s last comment about division however makes me think that he doesn’t have the same solution as me.

I think I got it, in about 2 minutes.

I will follow the lead of several others, stick my neck out, and state my belief that the puzzle as it stands has no solution. Looking forward to being proven wrong!

Having conceded defeat I went searching online for the solution, I take it back! I obtained the bulk of the solution but didn’t apply it, now I’m happy with the solution though. It is a very good puzzle.

I’ve not seen this before and after a minute I concluded that it was not possible. About five minutes later I figured it out. Nice puzzle!

one minute

Love the video – can you tell me the name of the song/musical group?

I have the solution that I believe is correct.

To me the real riddle is trying to find out the “flaw” in my answer…

I am doing some division, as Nick stated, but it is VERY simple division.

Simple along the lines of… One hour equals two half hours equals four fifteen minutes… Is this the division you’re referring to Nick?

There is AN assumption I can see, but in the context of a logic puzzle it is a reasonable one, and one that would be very straightforward to explicitely state in the question (and hence can hardly be said to “spoil the simplicity of the puzzle statement”). In saying that the rope burns end to end in an hour, it’s necessary to assume that it doesn’t matter

whichend you start from.Here’s an example where this assumption is violated. Suppose that, in minute N+1, either the same

volumeof rope is burned as was burned in minute N, or else the samelengthof rope is burned as was burned in minute N, whichever of those two options leads to the greater amount of rope being burned. Now, if you consider a rope that’s thick on one end and thin on the other, then it will burn out sooner if you light it on the thicker end.But honestly, this is a logic puzzle, not a physics experiment, and the assumption can be stated in just five words: It; Doesn’t; Matter; Which; End. I see no simplicity being spoiled here.

D’arcy, yes, that’s the division I’m referring to. I’m not suggesting you’re making an arithmetical error! I’m asking whether division is necessarily the appropriate mathematical operation to use in this case.

Flesh-eating Dragon, that’s not quite the assumption I’m thinking of. I agree it’s reasonable to assume, even with the existing wording, that it doesn’t matter which end of the rope you ignite. And, as you say, a simple clarification would not spoil the puzzle.

The assumption I have in mind would require a longer, more cumbersome clarification that would unfortunately also give a hint how to solve the puzzle.

Stop . . . wait for it . . . twisting yourselves into knots over this rope puzzle. No assumptions are violated in the volume/length/other department. The width not mattering is stated simply to indicate an unpredictable burn rate; Richard could just as well have stated the ropes were damp in places and would thus burn unevenly. All that matters is that the *total* length will burn in one hour; there is no “burn out sooner”. If you want to take issue with something, wait until Monday when everyone can discuss their full solutions.

The central premise of this puzzle is indeed the unpredictable, uneven burn rate. And that’s what seems to be catching you out. It simply isn’t true that, “All that matters is that the *total* length will burn in one hour…” There’s more to it than that.

Surprisingly… It took me 45 minutes!

While reading the puzzle (which I might have solved before but don’t consiously remember) I started pouring water between pitchers in my mind, and when I came to the “45 minutes” part I had a solution to match it already.

Reading through the comments I try to find this mentioned catch in my reasoning. But searching for hidden assumptions I can only come up with the one Flesh-eating Dragon mentions; that it would burn for one hour from end A to end B or equally from end B to end A. (Otherwise both ropes would have to be marked “Ignite here” or I wouldn’t know if it measured an hour or not.)

Well, I also assume a lot of stuff are irrelevant, like time to ignite or orientation of the ropes, etcetera.

I like how these comments make me doubt and scrutinize the solution I was prepared to take poison on… 🙂

Hope I’ve got it right! Took about 3 mins.

There is also a way for measuring 40 minutes with these ropes. Though you need to make some more assumptions as you do here, but it still stays a logical puzzle.

Looks like your website is under attack from supernatural forces…

http://isgodimaginary.com/forum/index.php/topic,40909.0.html

you really need to add comment moderation to your blasphemy…

Atlast! Got the answer in about a min. Pretty sure about the answer..

I think I find a solution (about 1 min)

Took me quite a while. I’d never heard this one before. I came up with three hypotheses that I was able to reject, and then the solution hit me, and it was so obvious. I think it took about 8-9 minutes.

OK, do you know why I can lift my 3rd finger? I’d think if I can, lots of people can.

Could you enlarge on this please?