Riddle me this

[Andail]

Ok, so we were a group of people and someone told a riddle that went something like this:

A man walks up a hill one day, following a trekking route. He begins at the foot of the hill at 7 am and reaches the top at 7 pm. The next day, he descends the hill, walking the exact same route, beginning at 7 am and finishing at 7 pm.
The route is naturally uneven, with steep, slow parts and more level, fast parts.

While descending the hill, will there be a point, anywhere, at any time, where he can say "Exactly at this time yesterday, I was here"?
In other words - is there sure to be a point that he reaches the same time both days?

I see myself as moderately intelligent - I did take some math courses in university, with probability and shit, but am far from a logical mastermind - but I had no problems whatsoever understanding this riddle, saying "yeah of course." However, almost nobody there, and almost nobody I've spoken about the riddle with, could see this solution.

Now, you guys are genuinely clever, so I'm turning to you. Is this riddle really that hard? Isn't the answer quite obvious - yes, there must exist such a point, regardless of his pace, and regardless of the terrain?

[Snarky]

Yes.

Spoiler

If we imagine that instead of going up one day and down the next, you have two people, one who's going down and one who's going up on the same day, there has to be a point where they pass each other, i.e. they're at the same position at the same time. The same thing applies when he's comparing himself with the time and position from the day before.

[close]

But I don't think it's obvious. My intuition would have been to say no, and I'm not sure I'd have seen it if you hadn't insisted the answer is yes.

[NickyNyce]

Pretty easy one...

What has forests with no trees, mountains with no rocks and cities with no people


and another...

What word starts with E and ends with E but has only one letter

[ThreeOhFour]

What word starts with E and ends with E but has only one letter

Spoiler

Envelope

[close]

[NickyNyce]

Well done ThreeOhFour

[Anian]

What has forests with no trees, mountains with no rocks and cities with no people

Spoiler

A map.

[close]

[Andorxor]

The answers is no,because the earth moves around the sun.

[Radiant]

The answers is no,because the earth moves around the sun.

That's Newtonian physics, and is outdated. By contemporary physics, it depends on your frame of reference.

[Kweepa]

Huh.
Like Snarky, my intuition said no until I read his solution.
It's amazing how reframing exactly the same puzzle can make the answer so obvious!

[Ryan Timothy B]

Isn't the answer quite obvious - yes, there must exist such a point, regardless of his pace, and regardless of the terrain?

As I read it I also found the answer to be obvious (unless you factor in what Andorxor said).

To get a little side tracked, going by what Andorxor said, I sometimes like to pick apart time travel movies that somehow end up on the same spot of the planet no matter the rotation and such. Even with the Delorean just traveling one minute into the future on its first test, it would have returned thousand miles away from its origin (or inside the planet depending where they were on earth rotation). Unless Doc had somehow managed to calculate the exact location and rotation of the earth (and sun, since our solar system and galaxy is shifting along) with any given time and create the return warp hole there. Then there's the moon slowing or altering our rotation, asteroids, etc; but it shouldn't matter too much in a 20 year span.


So my riddle (or question - whatever you'd call it lol): What does a clock measure?

[Anian]

As I read it I also found the answer to be obvious (unless you factor in what Andorxor said).

To get a little side tracked, going by what Andorxor said, I sometimes like to pick apart time travel movies that somehow end up on the same spot of the planet no matter the rotation and such. Even with the Delorean just traveling one minute into the future on its first test, it would have returned thousand miles away from its origin (or inside the planet depending where they were on earth rotation). Unless Doc had somehow managed to calculate the exact location and rotation of the earth (and sun, since our solar system and galaxy is shifting along) with any given time and create the return warp hole there. Then there's the moon slowing or altering our rotation, asteroids, etc; but it shouldn't matter too much in a 20 year span.

1. Well it's obvious unless you have a different thought process. Because, while they will meet at one time, the percentage of their trip will almost certainly not be the same, since the path is pretty random and undefined. Ok, no need to get further into it, I thought Snarky's solution was pretty clear and obvious once I've read it.

2. About the time travel. I have often thought about this very restriction. While some time travelling devices are also space travelling devices, a lot of them I think are not. It's really hard to talk about this because I think we (not only personally but as humanity) don't know enough about time travel in practical form and usage.
BUT I think you could make an interesting story setting out if it, because you could say that in entire time line there's a limited number of times that you could travel to that same spot and not "teleport" into empty space. Btw I think there was even Xkcd comic about it. Though again if you take into account that the universe is expanding and shifting all the time, you wouldn't actually be able to time travel anywhere, though time travel space ship would hav some cool calculations to make, so they don't travel into a meteor or something. Then again, I found that time travel stories are often best when time travel is a mechanic that is set by it's limits and it's explained how it works in the simplest or most general way and then get on with the story as the main thing that should be paid attention to (stuff like Time traveller's wife etc.)
Oh, let's not forget that f-ing brain twister that is the movie Primer.

[EchosofNezhyt]

However, almost nobody there, and almost nobody I've spoken about the riddle with, could see this solution.

Thats wierd.

[Crimson Wizard]

...I sometimes like to pick apart time travel movies that somehow end up on the same spot of the planet no matter the rotation and such. Even with the Delorean just traveling one minute into the future on its first test, it would have returned thousand miles away from its origin (or inside the planet depending where they were on earth rotation). Unless Doc had somehow managed to calculate the exact location and rotation of the earth (and sun, since our solar system and galaxy is shifting along) with any given time and create the return warp hole there. Then there's the moon slowing or altering our rotation, asteroids, etc; but it shouldn't matter too much in a 20 year span.

Here's the simple semi-scientific explanation: time travel is being made inside the gravity field of the planet, the time machine and the planet are bound system of bodies.

What does a clock measure?

It measures the amount of changes inside itself.

[Ryan Timothy B]

It measures the amount of changes inside itself.

The answer I liked was from the movie The Man from Earth:

Dan: Time... you can't see it, you can hear it, you can't weigh it, you can't... measure it in a laboratory. It is a subjective sense of... becoming, what we... are, instead of what we were a nanosecond ago, becoming what we will be in another nanosecond. The whole piece of time's a landscape existing, we form behind us and we move, we move through it... slice by slice.
Linda Murphy: Clocks measure time.
Dan: No, they measure themselves, the objective referee of a clock is another clock.

[Khris]

Cool thread!
I was intuitively sure that there had to be a point as described, but I couldn't say why until I read Snarky's brilliant explanation.

Re the clock, I'd say it doesn't measure anything, it just moves.

One of my favorite riddles: a brick weighs 2 kilos + half of its weight. How much does it weigh?

[Crimson Wizard]

One of my favorite riddles: a brick weighs 2 kilos + half of its weight. How much does it weigh?

Spoiler

Since the brick's weight consist of two halves, then 2 kilos is another half, and total is 4 kilos.

[close]

That's actually difficult only for those who think too hard

[Andail]

I think the trekking route riddle is interesting because it's such an everyday scenario, not like many other far-fetched riddles about monkeys and lightbulbs and stacks of coins and whatnot, and yet it's so hard for some to visualize the solution.

Perhaps I was only lucky to see the solution as Snarky described it - or actually, I kind of mentally superimposed the two routes, and saw that no matter how they're laid out, they have to cross at some point.

PS:
I see that Snarky moved this to the Rumpus forum, which I don't completely agree with, but I still hope it can steer clear from nonsense riddles en masse, and stay somewhat serious. I really like the time-travelling discussion, though

[Snarky]

Incidentally, I'm pretty sure this is a particular case of Brouwer's fixed-point theorem.

[Stupot]

It's not much of a hill - it takes the same amount of time to go up as it does to come down?
The only way this is possible is if you are walking at the same constant speed going down as you did coming up, and completely disregarding the differences in terrain.
If both sets of people are walking at the same speed and disregarding the differences evenness, then they will meet at 1pm (because they will have both been walking for 6 hours and both have 6 hours remaining of their journey.
Of course, this is crap, though, because we ARE taking into account the unevenness of the hill, so the two sets of people will probably not meet at 1pm.
The steep, slow parts going up will become the fast parts going down... the level parts will remain level on the return journey.
This is a hill we're talking about, so I can't understand what the man is doing on his way down, to still finish at 7pm. Maybe he stops off for a long lunch, in which case, any maths involved in this riddle can go out of the window.

[Edit]
To be honest, I think all the talk of it being a hill, and of steep/level slow/fast parts is just to throw you out. What matters is the fact that both journey's take 12 hours. Snarky's solution is perfect, but would just as well be the answer to a question about a man walking to the shop and back on a level road. I think for this riddle (and Snarky's response) to work, you have to ignore all the 'hill' stuff.  Maybe that is the point :/

[Andail]

Sorry, stupot, but you're incorrect. Apart from the fact that most hills are easier to descend than ascend. But the rest is still valid, and is what makes this riddle special. Otherwise the point would simply be in the middle.

[Atelier]

I also thought they would always meet in the middle, but this is assuming (a) the pace of man A is the pace of man B and (b) the path is completely flat - i.e. it's not a hill, but we know it is. But it can work, divide the hill into thirds with a steep first part, level second and steep third. On the way up the hill, the man's pace up the steep first part is slow so it takes him longer. In the same amount of time, the man walking down the hill has descended the steep part at the top and has also crossed the level part. The hill slopes in his favour so he walks faster, but he has longer to walk so they should arrive at the same time. A vague explanation but I think this is the principle?

But then surely the man walking uphill must speed up immediately afterwards, and the man walking downhill slows down his pace, so they both reach their destination at 7pm.

"The route is naturally uneven, with steep, slow parts and more level, fast parts."

This doesn't say which route we are talking about, whether uphill or downhill, because it's mentioned after both. What is steep for one is a decline for the other. If it is uneven then there could be steep parts going down on the path whilst you're moving up the hill. So on the way back down the hill, they would be steep going up, accounting for the slow in pace of B and quickening in pace of A.

[Stupot]

I don't see why it HAS to be a hill in the riddle.  The same applies to any journey (flat or otherwise). There's no point mentioning the slow/fast parts in the riddle if you're not going to take them into account (unless the purpose is to throw you off, which I think it is).

I agree the answer is that Yes, that there will be a point on the way back down where you could say that you had been at that exact spot at the exact same time the previous day (As Snarky neatly illustrates), but this is reliant totally on the fact that both journeys begin and end at the same time, and the terrain of the hill is completely irrelevent.  Unless of course you are trying to calculate exactly where and when that point would be, but that's not what the riddle is asking (Thank fuck!).

[Jonez]

Yeah, the terrain type and the length is irrelevant, but that's what makes it a riddle.

This was my approach to the answer, a bit like the two-persons-meeting -solution but maybe not as intuitive.

[Snarky]

It's not much of a hill - it takes the same amount of time to go up as it does to come down?
The only way this is possible is if you are walking at the same constant speed going down as you did coming up, and completely disregarding the differences in terrain.
If both sets of people are walking at the same speed and disregarding the differences evenness, then they will meet at 1pm (because they will have both been walking for 6 hours and both have 6 hours remaining of their journey.
Of course, this is crap, though, because we ARE taking into account the unevenness of the hill, so the two sets of people will probably not meet at 1pm.
The steep, slow parts going up will become the fast parts going down... the level parts will remain level on the return journey.
This is a hill we're talking about, so I can't understand what the man is doing on his way down, to still finish at 7pm. Maybe he stops off for a long lunch, in which case, any maths involved in this riddle can go out of the window.

Or maybe he twists his ankle when he's halfway down and has to go slower. It doesn't really matter. (By the way, walking downhill isn't necessarily faster than walking along a level path, particularly if it's slippery.) Yes, "maths" (by which I take it you mean our ability to calculate when and where he would be at the same place he was the day before) goes out the window. That's the point: If his speed is constant both ways, the answer is obvious. But what's the answer if it's not?

That's exactly the kind of question mathematicians are interested in, and the answer is a perfectly good (informal) non-constructive proof: Yes, we know that the point must exist, even if we can't say where it is (see the link I posted to Brouwer's fixed-point theorem). So mathematics as a whole is not out the window.

To be honest, I think all the talk of it being a hill, and of steep/level slow/fast parts is just to throw you out. What matters is the fact that both journey's take 12 hours. Snarky's solution is perfect, but would just as well be the answer to a question about a man walking to the shop and back on a level road. I think for this riddle (and Snarky's response) to work, you have to ignore all the 'hill' stuff.  Maybe that is the point :/

The variable rate of movement is meant to ask whether maybe, although he visits all the same points over the same length of time both days, he never visits the same point at the same time. It's questioning whether what holds true in the simple case also holds true in the more general case. So yes, it's very relevant. (The way the question is posed, it doesn't really matter that it takes exactly as long to walk down as to walk up. As long as he starts moving at the same time and only moves one way along the path, there's going to be such a point.)

That it's uphill and downhill isn't really relevant, except that it messes with our intuition and makes it easier to misinterpret the kind of graph Jonez drew (because you might think of the curves as just the uphill and downhill profiles, rather than as displacement by time, so you might think that maybe the curves don't "reach" the intersection point at the same time).

Anyway...

How about the burning rope riddle? If you don't know it, it goes like this:

Imagine you have two ropes, which work like fuses. They'll each burn in exactly one hour. But the rate at which they burn is not constant (they won't necessarily have burned halfway in half the time; it might take 59 minutes to burn half the length, then only 1 minute to burn the rest). How can you use them to measure a period of 45 minutes?

[Crimson Wizard]

Imagine you have two ropes, which work like fuses. They'll each burn in exactly one hour. But the rate at which they burn is not constant (they won't necessarily have burned halfway in half the time; it might take 59 minutes to burn half the length, then only 1 minute to burn the rest). How can you use them to measure a period of 45 minutes?

That sounds as an extended version of one-rope riddle.

Spoiler

Put the first rope on fire from both ends and the second rope from one end. First rope will burn in 30 minutes (hour / 2), the second has still 30 minutes to burn. Immediately put the other second rope's end on fire. It will now burn in 15 minutes.
45 minutes total.

[close]

[Stupot]

@ Snarky - Yeah, I get it now.  In fact it's so obvious that I was just overthinking it. I was just thrown off by the fact it was a hill and the assumption that going down would be quicker than going up, when really it's so much simpler than that, because the same phenomenon applies to absolutely any journey.  It's actually very interesting that it is impossible to avoid such a point.  I made a little diagram that illustrates this.

Code: AGS

     0   1  2     3 4   5  6 7   8     9   10     11 12
Man A|---|--|-----|-|---|--|-|---|-----|---|------|--|
Man B|----|-----|-----|-|-|-----|----|------|--|-|-|-|
     12   11    10    9 8 7 *   6    5      4  3 2 1 0

If it was a flat journey the lines would be at regular intervals and the points would meet exactly at the end of the 6th hour (1pm), but as this is a journey with irregular terrain, the lines are all over the place, but there is still a point somewhere between the 6th and 7th hour where both timelines meet.

It was interesting when you said "it doesn't really matter that it takes exactly as long to walk down as to walk up."  As you say, as long as they start at the same time, it doesnt really matter when they finish.

Code: AGS

     0   1  2     3 4   5  6 7   8     9   10     11 12
Man A|---|--|-----|-|---|--|-|---|-----|---|------|--|
Man B|------|----|-------|---------|----------|------|
     6      5    4 *     3         2          1      0

Even if it takes half the time for Man B to descend the hill, the same remains true. In this case that time is some point between the 3rd and 4th hour. I was sure I could break this, but I couldn't haha.

Really, it seems so obvious now, that I feel like I've wasted time by thinking about it too hard. Haha.

As for the Rope one, I know this one and it's a bitch.  A friend told a group of us on a bus trip and nobody could work it out until he explained it at the end of the trip.

[Creamy]

Spoiler

I don't see how you can be sure that the rope will burn in 30 minutes if you put both ends on fire.

[close]

Anyway, what about this one : " the princess is half younger than the prince was when the princess was three times younger than the prince is. How old are they? "

Sorry if the grammar is bad.

_{edit by Andail: Please use hide tags when discussing the solution of a riddle}

[Andail]

Glad you're agreeing with us now, Stupot I think Jonez's illustration is also very pedagogical.

I like the rope riddle, it has some things in common with the first one, actually.

Creamy:

Spoiler

I don't see how you can be sure that the rope will burn in 30 minutes if you put both ends on fire.

Spoiler

If you know that the entire rope will burn in one hour, then lighting both ends simultanuously will solve the task of finding the "true" middle of the fuse. Both sparks will find this point after exactly 30 mins, and then there will be no more rope, and the fire will go out.

[close]

[close]

[Crimson Wizard]

Creamy, regarding ropes

Spoiler

I don't see how you can be sure that the rope will burn in 30 minutes if you put both ends on fire.

No matter how fast each piece of the rope burns, the fire will have to "pass" every one of them to burn whole rope. If fire goes from both ends rope will burn x2 faster.
Neither length of the rope, nor flammability of each piece matters, it is only overall burning time that does.
We may take this analogue: imagine there are X items on the floor, each of different weight and it takes different time to pick each up and put it in the crate. Now, isn't that obvious that two men will move them to the crate x2 faster that one? (Here we make a simplification that they are of same strength and do not get tired)

[close]

Anyway, what about this one : " the princess is half younger than the prince was when the princess was three times younger than the prince is. How old are they? "

Now, this is a funny case. I was totally sure there's some tricky solution and certain answer. But then I decided go math, and found... something different.
There's a chance I made some dumb mistake, but I got this:

Spoiler

Code: ags

X0; // Princess age before
Y0; // Prince age before
X1; // Princess age now
Y1; // Prince age now

X1 = Y0 / 2;       // the princess is half younger than the prince was
X0 = Y1 / 3;       // the princess was three times younger than the prince is
// Or...
Y0 = 2 * X1;
Y1 = 3 * X0;

// We, of course, understand that the difference between their ages is always the same
Y1 - X1 = Y0 - X0;

// Hence...
3 * X0 - X1 = 2 * X1 - X0;
3 * X0 + X0 = 2 * X1 + X1;
4 * X0 = 3 * X1;
X0 = 3/4 * X1;

X0 = 3/4 * (Y0 / 2);
X0 = 3/8 * Y0;

Now, this means we may choose practically ANY Y0 value, calculate X0 from it and still got right answer.
For example:

Prince's age was Y0 = 100;
Princess' age was X0 = 3/8 * 100 = 37.5;
Prince's age now Y1 = 3 * X0 = 112.5;
Princess' age now X1 = Y0 / 2 = 50;

Now, checking this.
The princess is half younger than the prince was: Prince was 100, she is 50 now (half younger), correct.
...the prince was when the princess was three times younger than the prince is: Princess was 37.5, Prince now is 112.5 (three times older), correct.
Time passed for prince: 112.5 - 100 = 12.5 years, time passed for Princess: 50 - 37.5 = 12.5, the same, correct.

Now, tell me what did I wrong? 

[close]

[Snarky]

Cool Stu!

Spoiler

I don't see how you can be sure that the rope will burn in 30 minutes if you put both ends on fire.

[close]

If Andail's or CW's explanations don't convince you, you can think of it this way:

Spoiler

If the whole rope burns in 60 minutes if you burn it from one end, the part that hasn't burned after the first 30 minutes will have to burn in the remaining 30 minutes. In other words, the part that burned took 30 minutes to burn, and the remaining part will take 30 minutes to burn. So if you do burn it from both ends, after 30 minutes both parts will have burned.

[close]

[Creamy]

regarding ropes : I get it. I think I was misleaded by the fact that one end burns faster than the other.

To Crimson Wizard, regarding prince & princess ages :

Spoiler

Exact! You can find many solutions.
E.g. :                formerly          now
           princess    12               16
           prince      32               36
           
           princess    24               32
           prince      64               72

And so on...

[close]

[2ma2]

Yeah, the terrain type and the length is irrelevant, but that's what makes it a riddle.

This was my approach to the answer, a bit like the two-persons-meeting -solution but maybe not as intuitive.

This nails it. Independant on the shape of each graph, if Y equals distance from the starting point and X the time traversed each day, the graphs HAVE to intersect at some point.

Incidently, the brick puzzle is also easily explained if we abstract it:

• 2 + x = 2x
• x = 2

Now to dwell into the other riddles!

[Crimson Wizard]

• 2 + x = 2x
• x = 2

I'd say it is more mathematically correct to write 2 + x/2 = x, since it is total weight of brick we are looking for, not half of weight.

[Andail]

Hi Linus

[2ma2]

'lo Petter!