[Guest]

Chuck Norris decided he needed a breather from protecting the world, so he took a two day trip to a mountain. He reached the base of the mountain at 8 am the first day, and proceeded to run at a fast pace all the way to the top, pausing only briefly to move some large boulders off the trail. He reached the summit at 5 pm, put down a couple cold ones he had brought with, then spent the night comfortably contemplating the stars.

The following morning he returned down the mountain along the same trail he had climbed, again departing at 8 am. With gravity on his side (not that he needed its help), he practically flew down the mountain. However, as a Texas Ranger with a highly developed appreciation of nature, he couldn't help but stop several times along the way to quietly observe his beautiful surroundings. Please note that he did not stop because he was out of breath. In any case, utilizing this careful blend of sprinting and pausing, he reached the base at 1 pm.

Now the question: Given that he did not actually leave the trail at any point while traveling, how can you prove that Mr. Norris was at the same place on the trail at the same time of day on both days?

[Guest]

Simple math, you'd need the speed and the distance though.

[Guest]

Simple math, you'd need the speed and the distance though.

You shouldn't need any more details than what I've provided.

[Guest]

Well as we only have to prove it, I was watching and recorded it.

[Guest]

Not really "proof" in the mathematical sense of the word, but I'd be inclined to plot the movements of Chuck (we're on 1st name terms), as a graph of "time" vs. "where on trail". One line starts at the bottom at 8am and reaches the top at 5pm. Another starts at the top at 8am and reaches the bottom at 1pm. The two lines must cross at some point. That would be the point where he was at the same place at the same time.

However, there is a flaw in this logic. It is quite possible that Chuck may have chosen to descend at such a speed that his motion became discontinuous (he travelled from A to B without passing through any point in between). Chuck can do this, I saw it on telly (or it might have been dodgy editing). If so he may have avoided all attempts at applying logic to his motion. Never underestimate the power of Chuck.

[Guest]

Not really "proof" in the mathematical sense of the word, but I'd be inclined to plot the movements of Chuck (we're on 1st name terms), as a graph of "time" vs. "where on trail". One line starts at the bottom at 8am and reaches the top at 5pm. Another starts at the top at 8am and reaches the bottom at 1pm. The two lines must cross at some point. That would be the point where he was at the same place at the same time.

However, there is a flaw in this logic. It is quite possible that Chuck may have chosen to descend at such a speed that his motion became discontinuous (he travelled from A to B without passing through any point in between). Chuck can do this, I saw it on telly (or it might have been dodgy editing). If so he may have avoided all attempts at applying logic to his motion. Never underestimate the power of Chuck.

I agree with the possibilities of the aforementioned solution but add that it is also possible that Chuck was in a commercial break during the time when his upward and downward paths would have crossed. Therefore unless the commercial was for a Total Gym, Chuck would have temporarily ceased to exist thereby missing his opportunity to be at the same point at the same time. He would have resumed the path, at his convenience, after the break.

[Guest]

Not really "proof" in the mathematical sense of the word, but I'd be inclined to plot the movements of Chuck (we're on 1st name terms), as a graph of "time" vs. "where on trail". One line starts at the bottom at 8am and reaches the top at 5pm. Another starts at the top at 8am and reaches the bottom at 1pm. The two lines must cross at some point. That would be the point where he was at the same place at the same time.

Yes, it really is that simple. However, when I first read a variation of this a few years ago, it took me a while to get my head around it, even after I understood the mathematical premise. Another explanation is to visualize the mountain with Chuck at the bottom and at the top, leaving at the same time. At some point they have to cross on the trail. It's the same thing as what you said, just less mathy.

[Guest]

Yes, it really is that simple. However, when I first read a variation of this a few years ago, it took me a while to get my head around it, even after I understood the mathematical premise. Another explanation is to visualize the mountain with Chuck at the bottom and at the top, leaving at the same time. At some point they have to cross on the trail. It's the same thing as what you said, just less mathy.

That particular visualisation leads to a paradox. If Chuck met himself coming down, he would have to have a fight with himself to decide who can continue on the path. In order for one Chuck to win the fight, the other must lose. Chuck losing a fight is logically impossible and would cause a rift in the space-time continuum. Hopefully Chuck would be able to jump over the rift and continue on the path without it troubling him too much.

[Guest]

The wording of the question is a little ambiguous. Do you want to prove where and what time he was at the same location, or just that he had to be somewhere at the same time both days?

One possibility is that Chuck started up at 8:00am and one minute later started moving boulders for 5 hours until 1:01pm then climbed the rest of the way. On the way down, he would at some point cross the point where he was moving boulders, say at perhaps 12:59pm. Thus he was at that same point at 12:59pm on both days.

There are infinite possibilties though, so to prove the exact location and time requires more info, but yes, he must cross somewhere at the same time.

[Guest]

Chuck Norris has the ability to be anywhere whenever he wants. Its been documented that he once delivered a baby in Kansas while also killing Mexican drug smugglers with a series of round house kicks. The laws of physics don't apply. He only obeys one law...the law of Chuck Norris.

[Guest]

The wording of the question is a little ambiguous. Do you want to prove where and what time he was at the same location, or just that he had to be somewhere at the same time both days?

One possibility is that Chuck started up at 8:00am and one minute later started moving boulders for 5 hours until 1:01pm then climbed the rest of the way. On the way down, he would at some point cross the point where he was moving boulders, say at perhaps 12:59pm. Thus he was at that same point at 12:59pm on both days.

There are infinite possibilties though, so to prove the exact location and time requires more info, but yes, he must cross somewhere at the same time.

It's not ambiguous. It just asks for the simpler of the two possibilities you suggested.

It's not at all hard to prove, but when I first encountered it, it seemed to me there had to be some way that he could avoid being somewhere on the trail at the exact same time on both days. In retrospect, Chuck Norris was probably not the best person to use for this puzzle, since he might be the only human capable of escaping the usual limitations of the space-time continuum.

[Guest]

Needing a way to track his path of devastation, Chuck created CPS, the Chuck Positioning System. Once perfected, he opted for anonymity, and changed the name to GPS. If you can speak in a headlock, you could just ask Chuck for the answer...but beware of the roundhouse kick.

[Guest]

Chuck Norris decided he needed a breather from protecting the world, so he took a two day trip to a mountain. He reached the base of the mountain at 8 am the first day, and proceeded to run at a fast pace all the way to the top, pausing only briefly to move some large boulders off the trail. He reached the summit at 5 pm, put down a couple cold ones he had brought with, then spent the night comfortably contemplating the stars.

The following morning he returned down the mountain along the same trail he had climbed, again departing at 8 am. With gravity on his side (not that he needed its help), he practically flew down the mountain. However, as a Texas Ranger with a highly developed appreciation of nature, he couldn't help but stop several times along the way to quietly observe his beautiful surroundings. Please note that he did not stop because he was out of breath. In any case, utilizing this careful blend of sprinting and pausing, he reached the base at 1 pm.

Now the question: Given that he did not actually leave the trail at any point while traveling, how can you prove that Mr. Norris was at the same place on the trail at the same time of day on both days?

With math. I got 11:13 am and about 15 secs. I used fractions and trial and error and narrowed it down. Although an equation could probably be made. Nice brain buster.

[Guest]

Yes, it really is that simple. However, when I first read a variation of this a few years ago, it took me a while to get my head around it, even after I understood the mathematical premise. Another explanation is to visualize the mountain with Chuck at the bottom and at the top, leaving at the same time. At some point they have to cross on the trail. It's the same thing as what you said, just less mathy.

I'm still not convinced...I don't see how that is necessarily true because the 'paths' move at variable speeds...I'm with you in that the up and down paths must cross at some point, but because the rate of travel varies, I don't see how you can say that, when the lines cross, it will be at the same time...The 'same place' might be at 11AM on the way up, but at 9AM on the way down, because the rate of travel (up and down) is not constant...correct? It would be fairly straightforward if the rate of travel is constant, but since it varies in ways unkown, I can't visualize how you could make any declarative statements about WHEN something happens...

Smoo

Edited May 21, 2008 by Smoo

[Guest]

Chuck Norris decided he needed a breather from protecting the world, so he took a two day trip to a mountain. He reached the base of the mountain at 8 am the first day, and proceeded to run at a fast pace all the way to the top, pausing only briefly to move some large boulders off the trail. He reached the summit at 5 pm, put down a couple cold ones he had brought with, then spent the night comfortably contemplating the stars.

The following morning he returned down the mountain along the same trail he had climbed, again departing at 8 am. With gravity on his side (not that he needed its help), he practically flew down the mountain. However, as a Texas Ranger with a highly developed appreciation of nature, he couldn't help but stop several times along the way to quietly observe his beautiful surroundings. Please note that he did not stop because he was out of breath. In any case, utilizing this careful blend of sprinting and pausing, he reached the base at 1 pm.

Now the question: Given that he did not actually leave the trail at any point while traveling, how can you prove that Mr. Norris was at the same place on the trail at the same time of day on both days?

Wow I was the first to actually give a time. Although I did assume that good ol Chuck moved at a constant rate going up and a differnt constant rate going down. You would need a steady rate of travel to figure this out.

[Guest]

I'm still not convinced...I don't see how that is necessarily true because the 'paths' move at variable speeds...I'm with you in that the up and down paths must cross at some point, but because the rate of travel varies, I don't see how you can say that, when the lines cross, it will be at the same time...The 'same place' might be at 11AM on the way up, but at 9AM on the way down, because the rate of travel (up and down) is not constant...correct? It would be fairly straightforward if the rate of travel is constant, but since it varies in ways unkown, I can't visualize how you could make any declarative statements about WHEN something happens...

Smoo

I think that because he pauses only "breifly" we can assume a constant rate of travel. But you know what they say about assuming.

[Guest]

Because of the unspecified-length pauses, you can't find a unique solution to the problem. You can non-constructively prove the existence of a solution, though.

1. Let x(t) be his height at time t during the ascent, and y(t) be his height at time t during the descent. Let time t=0 be 8AM for simplicity, with t measured in hours. x(t) and y(t) must be continuous, since he moves with finite speed.

2. Let d(t) be y(t)-x(t). d(t) must be continuous since x(t) and y(t) are continuous.

3. d(0)=H, where H is the height of the mountain.

4. d(5)<0, since y(5)=0 and x(5)>0.

5. By the intermediate value theorem, there must exist some time c in the interval [0,5] s.t. d©=0.

QED

[Guest]

Now the question: Given that he did not actually leave the trail at any point while traveling, how can you prove that Mr. Norris was at the same place on the trail at the same time of day on both days?

The answer is really quite simple: When he reached the base of the mountain on the first day, Chuck roundhouse kicked it in what he presumed to be its face. As usual, the force was so great that it created a rift in space-time, as evidenced by the fact that the event was observed in slow motion three times in succession. By the time the cosmic dust settled, it was 1PM the following day in the earth's frame of reference. Meanwhile, Chuck had gotten 3 years younger, and his beard had found a cure for cancer.

[Guest]

I'm still not convinced...I don't see how that is necessarily true because the 'paths' move at variable speeds...I'm with you in that the up and down paths must cross at some point, but because the rate of travel varies, I don't see how you can say that, when the lines cross, it will be at the same time...The 'same place' might be at 11AM on the way up, but at 9AM on the way down, because the rate of travel (up and down) is not constant...correct? It would be fairly straightforward if the rate of travel is constant, but since it varies in ways unkown, I can't visualize how you could make any declarative statements about WHEN something happens...

Smoo

I think you are being fooled by the simplicity of the problem. I think you're trying to bring a third dimension into the problem. There are only two dimensions: position along the path and time. You mention that the 'same place' could be at 11AM up 9AM down, which could be true. But there are an infinite number of 'same places' since his entire path is the same. So using your example, there could be another 'same place' at 10:59AM up and 9:01AM down, right? Then perhaps there is another 'same place' at 10:30AM up and 9:30AM down, and yet another 'same place' at 10:15AM up and 9:45AM down. So for your example, it is possible that he crosses the 'same place' at 10:00AM up and 10:00AM down, right?

You said "the up and down paths must cross at some point", but the up and down paths actually cross at every point, not just some point.

Hope that helps and doesn't confuse things.

Edited May 21, 2008 by MAD

[Guest]

I'm still not convinced...I don't see how that is necessarily true because the 'paths' move at variable speeds...I'm with you in that the up and down paths must cross at some point, but because the rate of travel varies, I don't see how you can say that, when the lines cross, it will be at the same time...The 'same place' might be at 11AM on the way up, but at 9AM on the way down, because the rate of travel (up and down) is not constant...correct? It would be fairly straightforward if the rate of travel is constant, but since it varies in ways unkown, I can't visualize how you could make any declarative statements about WHEN something happens...

Smoo

Another way to think of it is this:

It's the first day, and Chuck Norris's beard, the source of his power, has climbed to the top before him. At 8am, Chuck begins sprinting up the trail, and his beard sprints down. At some point they have to meet or pass each other. That is when they are at the same place at the same time, even if his beard is WAY faster than the rest of him, and even if each one pauses periodically. Because of the pauses, you can't use math to figure WHERE that place and time is, but you know it exists.

[Guest]

I'm still not convinced...I don't see how that is necessarily true because the 'paths' move at variable speeds...I'm with you in that the up and down paths must cross at some point, but because the rate of travel varies, I don't see how you can say that, when the lines cross, it will be at the same time...The 'same place' might be at 11AM on the way up, but at 9AM on the way down, because the rate of travel (up and down) is not constant...correct? It would be fairly straightforward if the rate of travel is constant, but since it varies in ways unkown, I can't visualize how you could make any declarative statements about WHEN something happens...

Smoo

If you're using the line graph method, where one axis is time and the other is position, then when the lines cross, they will both have the same time and position. I'm not sure what MAD meant, but there will only be one time/position on the trail where this is the case.

The problem states that Mr. Norris pauses on the way up and the way down, so we know his rate of ascent and descent are not constant. Therefore, while we can't say exactly when or where his time and position will be the same, we can be sure that at some point they will be.

Edited May 22, 2008 by Duh Puck

[Guest]

What graph is used... position vs time, or distance vs time? With distance I think it's impossible, but position/displacement makes more sense... also, i have no idea how people are getting mathematical values. you have no idea what times chuck stopped at. So your values would be innaccurate, correct?

[bonanova]

I did not do the math, since Chuck Norris never uses math when he shaves.

However, I did sketch a plot of the entire event, introducing zigs and zags

at the precise moments of his boulder movement and nature gazing, by

channeling his motions as I read the accounts.

Thus I was able to determine that the exact place on the trail where he met

himself coming down was 0.6832" +/- 0.0003" from the left edge of the sketch.

I will scan and post the sketch if there are doubters.

[Guest]

I will scan and post the sketch if there are doubters.

I don't believe you.

Actually, I'd just love to see your sketch.

[Guest]

I did not do the math, since Chuck Norris never uses math when he shaves.

However, I did sketch a plot of the entire event, introducing zigs and zags

at the precise moments of his boulder movement and nature gazing, by

channeling his motions as I read the accounts.

Thus I was able to determine that the exact place on the trail where he met

himself coming down was 0.6832" +/- 0.0003" from the left edge of the sketch.

I will scan and post the sketch if there are doubters.

+/- 3 tenths? What did you use to measure to measure that, an electron microscope?

[Guest]

That particular visualisation leads to a paradox. If Chuck met himself coming down, he would have to have a fight with himself to decide who can continue on the path. In order for one Chuck to win the fight, the other must lose. Chuck losing a fight is logically impossible and would cause a rift in the space-time continuum. Hopefully Chuck would be able to jump over the rift and continue on the path without it troubling him too much.

- Have to mention that BL beat CN. It's true and it was coaught on film CN died, Did he run up/down the mountain before that?

9 hours up and 5 hours down

2/3 the way down?