# What colour was the.... No, there's no damn bears here [closed]

Puzzlers of the world were gathering at the PuzzleMania conference-as a special event, celebrities from various fields of endeavour were invited to give their favourite puzzles.

It came to Ed the Explorer's turn:

This one will make you really think. Once, whilst exploring, me and my men headed for 100 units due south, then headed for 100 units due east and then headed for 100 units due north, and we ended up exactly where we had started. How is that possible?

But by then, there was a ripple going around the room. As soon as they had heard the first sentence, there were mutterings...

Oh not that one again-how many times have we heard this? Can't he come up with something original... Bo-ring..

It has nothing to do with the North Pole... or poles of any sort... or anything to do with bears of any colour

So then where were Ed and his men and how did they manage to do this?

To address some ideas the puzzlers at the conference had, Ed would like to clarify:

1. They did actually travel the distances stated
2. The distances they traveled were in a straight line so when they went 100 units east, they ended up 100 units east of where they started and didn't just go around in circles.
3. When they had completed the 100 units north, they were back exactly where they started.
• Well, there goes my merry-go-round theory, as well as my giant hamster ball hypothesis. Feb 5 '18 at 21:12
• @gnovice and my Google Maps theory as well. Feb 5 '18 at 21:34
• @Alconja I'm pretty sure he means Ed the Explorer, not Seyed Feb 6 '18 at 7:31
• Voting to reopen, since every single answer (I checked) on the linked dupe is explicitly disallowed by one or more of the constraints here.
– Bass
Feb 6 '18 at 10:01
• That was a fun and interesting puzzle, but after reading your own answer, I must sadly conclude that it is indeed too broad. There are simply too many possible answers for that and finding the correct one ends being a mind-reading guessing game with a hopelessly large and unverifiable space for searching. Feb 7 '18 at 18:07

One answer without any circles whatsoever is that they were

On a ship, like explorers often are

This works, because

The heading of a ship isn’t the only factor in deciding the actual direction of movement, you have to factor in the wind pushing you sideways, and water currents too.

So in their own reference frame they travelled in a sort of rectangular U, but

while doing so, they drifted 100 units west, so their path formed an equilateral triangle. Like so:

To achieve this exact path shape, the magnitude of the westward drift needs to be $\frac{1}{\sqrt{3}} \approx 0.577$ times the ship's speed in still water.

This fits the other clues too, since at all times, they were actually moving in straight lines, every part of the trip moved them 100 units, and

during the eastward portion of the trip, the drift only slowed them down, not causing any directional deviation, so they actually did end up exactly 100 units east.

• I think your third and fourth hidden statements are contradicting each other. If we assume the fourth, well, your whole idea doesn't work anymore. Feb 6 '18 at 14:50
• Please elaborate? If they headed south in a constant current that dragged them westward at $\frac{1}{\sqrt{3}}$ of the ship’s still-water-speed, and changed heading after traveling 100 units, like the problem states, then surely the resulting path would be an equilateral triangle, leaving them in the exact same spot where they started. Certainly, the north-south portions of the trip won’t leave them 100 units north/south of the starting point, but exactly that requirement is not given, just that the trip leaves them 100 units away.
– Bass
Feb 6 '18 at 15:18
• @Evargalo, I think it checks out. OP clarified that when they went 100 units east, they ended up 100 units east of where they started, but didn't actually stipulate anything similar for the north/south axis. They could be "heading" N but actually traveling NW in roughly a straight line. No reason to assume current/wind conditions would be the same on the way out and on the way back (I suspect this was just an omission by the OP, and they meant what they said about eastward travel for both axes, but so what). Feb 6 '18 at 15:19
• @Guest Well, one interesting feature of this solution is that it would work on a flat earth too :-)
– Bass
Feb 6 '18 at 19:21
• @Guest They need to keep moving eastward. The current is obviously flowing towards the edge; if they hit the event horizon there's no direction that leads away from the edge. Feb 6 '18 at 22:00

While the weasel word "units" is tempting to pick at, I think there's a much more mundane solution, that doesn't involve redefining words or strange time-space geometries.

The puzzle specifies what Ed the Explorer and his men did, but

not what they didn't do. So, it's quite possible that something happened in between one of their steps.

For example:

Ed the Explorer (and his inexplicably all-male crew) are deep sea explorers hunting for wondrous (and potentially cursed) treasure. They travel 100 units south to where they think they'll find a fabled, sunken pirate ship. But disaster strikes! Out of nowhere a sudden storm overwhelms their ship, capsizing it. Even if he was conscious, there is nothing Ed could do in such a storm. But seeing as he inhaled quite a bit of salt water, Ed is not conscious. When he comes to, the first thing he sees are eyes looking back at him. Beautiful, native eyes. Hurray! He's been rescued. What's more, the natives have completely repaired his ship. The storm has past, so Ed the Explorer and his team of manly men immediately head back to the location of the pirate ship, which happens to be due East, 100 units. The water is so clear, Ed thinks he can see the twinkling of doubloons far below. He is joyfully hopping foot-to-foot until his first mate, shouts, "Look to the sky! It's the curse!" Ed looks up and recognizes at the horizon the dark red tinge that warns of a coming squall. "We'll risk it, men! For are we not men? Yes, we are men! All of us. Men." Stirred by this rousing speech, Ed the Explorer's men begin preparations for hauling the pirate booty up, but no sooner has the first dinghy touched the water when the storm arrives with a ferocious torrent. Ed the Explorer decides at that point that, while it is a man's duty to stare nature in the eye and spit, he would prefer to remain a man and not become, say, food for the fish. Ed gives the order: "Head due north, 100 units! We're going back where we started, we're going home."

• Would +1 more than once if I could, both for the plausible answer and the brilliant narration Feb 6 '18 at 14:24
• Good thing you can't. Feb 6 '18 at 15:27

This can happen anywhere near the south pole, I hope this picture will explain the solution:

• I believe this violates the condition that they ended up 100 units east and did not go in circles. Feb 5 '18 at 23:05
• @HughMeyers To be fair, they didn't go in circles. They went in circle. Feb 6 '18 at 3:13
• Note that this is just one of many solutions of this type, where the 100 units equals an integer number of times around, as is already mentioned on the proposed duplicate. Although, I, also, would consider this solution precluded by the statement that Ed the Explorer didn't go in circles. Although, there is the possibility that there is an intended semantic difference between "circles" and "a circle". Feb 6 '18 at 5:24
• This definitely has to do with poles of some kind (the south kind), and is very much going around in circles: walking near the south pole, you have to keep turning right if you want to keep going east.
– Bass
Feb 6 '18 at 7:17
• "The distances they traveled were in a straight line." The eastward travel is not a straight line, as a "line" on the surface of a sphere is a great circle. Feb 6 '18 at 16:05

Were they...

at an amusement park? They could have walked 100 units south, gotten on a merry-go-round, walked east on the merry-go-round at the same speed that the merry-go-round rotated counter-clockwise for a total of 100 units, then gotten off and walked 100 units north to where they started.

Clearly,

The units mentioned are units of time. The group faced south for 100 minutes, then faced east for 100 min, then north for 100 min, and when they were done they hadn't moved!

Alternatively,

The explorers were lost. They walked south 100 m, east 100 m, and north 100 m, and when they were done, they were still lost.

• The first comment on the question by OP (now incorporated into the puzzle itself as an edit) contraindicates both of your alternative solutions. Good try though!
– Rubio
Feb 6 '18 at 7:35
• Rubio's right that these can't be the answer (anymore), but that second one had me laughing. :-D Feb 6 '18 at 15:17

A pretty obvious answer that I haven't seen anyone posting:

The size of the units is 0. Go 100 units south, and you haven't moved Go 100 units east, and you still haven't moved Go 100 units north, and you still haven't moved Throughout the whole 'journey', the team hasn't moved and so, end up in the same place.

• Well, I suggested 1mm units. The idea was to have the units be a "vanishingly small" distance. But hey, 0, that's different and unique. Feb 6 '18 at 15:55
• Except that then they wouldn't be traveling in straight lines. Feb 7 '18 at 16:59

there are 4 different men in the description, one called "me and my men", two called "then" and a final person called "we"

looking at the text again

Once, whilst exploring, me and my men headed for 100 units due south, then headed for 100 units due east and then headed for 100 units due north, and we ended up exactly where we had started. How is that possible?

So

'me and my men', 'then' and 'then' all travelled 100 units in some direction.

but obviously

'we' never moved so was exactly where s/he started

• In fact, 'me and my men' traveled 100 units. But 'then' traveled 200 units in an right-angled polyline. This is similar, but not the same as you wrote in the third yellow block. Anyway, a nice answer! Feb 7 '18 at 14:40
• Its also a possibility that Me and my men was literal - and the only named person was "we" - who has been ditched at the old camp 100 units west of the parties new location.. poor guy Feb 7 '18 at 15:54

could it be that

the circumfernce of the globe is 100 units and they are on the equator

• Welcome to Puzzling! go ahead and take the tour to earn your first badge. Feb 5 '18 at 23:32
• Could you please explain how this works (how you believe it works)? Please do not respond in comments; edit your answer to make it clearer and more complete. Feb 6 '18 at 1:46
• It couldn't. For that to work they'd have to travel the same distance south first, but they can't travel that far due south; they can only travel half that distance before they'd have to head north again (because they'd be at the south pole halfway through... not to mention "no poles involved"). Feb 6 '18 at 15:46

I got fed up of the "go in a circle, any circle" solutions, so here is a genuinely and completely novel solution for you..!!!

Have a think about possible ways before clicking the spoiler if so inclined :)

If they are on a world which itself rotates, and the axis of rotation is not aligned with the magnetic axis but at the right angle to it (you'd have to calculate the details, there will be a whole family of solutions)..... Ie as they walk, the _ poles themselves move in a great circle southward and around, then north, south and east will gradually change relative directions as they walk (so they will walk in arcs not straight lines). It would give rise to a while series of integrals for where they end up, with a whole series of related solutions in which walking speed, world size, rotation axis and speed, combine to produce viable solutions. Heading for food so no time to do the actual maths!! :)

• In the non-puzzling world, the words "north", "south, and "east" (and "west") refer to geographic directions based on the axis of rotation, not magnetic directions, unless otherwise specified.
– Mark
Feb 5 '18 at 23:51
• They might. Or they might not. We can't know this, since we have all lived (and developed our way of thinking) on a world where for all practical purposes the two coincide. We have no way to know, on a world where rotation is much more of a feature, whether we would still do so. (Or as another example suppose you were on a small moonlet like Phobos. Small moonlet, arbitrary rotation, just large enough for cardinal directions to be useful. Maybe for ease 'North' would be shorthand for 'whichever way the magnetic field says".) Its definitely capable of solutionhood and not precluded. I like it. Feb 6 '18 at 2:18
• Interesting idea, but perhaps a bit too creative to be the intended solution. While I agree that "North" has multiple meanings, magnetic north being one of them, it'd be nice if there was an actually existent, rather than hypothetical, planet in which "North" meant what you're suggesting. (Also: Good point about Phobos, but, once we get there, I don't think we're going to be relying on compasses to navigate. Magnetic fields are messy.) Feb 6 '18 at 12:22
• Now that I think about it, you know, your idea doesn't have to go to the extreme of putting the magnetic poles on the equator. The Earth's magnetic north pole is wandering somewhere around 500km (the magnetic inclination) away from the true north pole. That might be enough for you to try your fancy integrals on it. Feb 6 '18 at 12:35
• I'm taking this as a moral victory - the day when a solution here is "too creative"... Heaven forbid ;-) Feb 6 '18 at 12:36

Could they be

in space?

As such

When they have moved 100 units south/east/north the world has also rotated beneath them, as such, they are in the same position as they originally started.

• This occurred to me too, but wouldn't they be using some other coordinate system? Feb 6 '18 at 19:47
• @Guest I was considering a system where even though the ship has moved in the direction requested at the units requested, it still remains in the same position relative to an observer below and as such it it's in the same position.
– Kami
Feb 7 '18 at 11:27
• I get what you mean, but would you use terrestrial coordinates line "X units east, Y units north" in space? I'm not sure space travelers would say they traveled "X units east." Feb 7 '18 at 11:30
• This was I had in mind-Ed is a space explorer on the International Space Station.Say that a 'unit' is 100 miles.The ISS travels at c.15,000 mph so moving 100 units south(10,000 miles) would take approximately 40 minutes,then the same for due east and north.If the time taken for the ISS to make this U-shaped journey could be timed to coincide with the time of the Earth's orbit(plus the time to revolve an extra 10,000 miles),then when the station had completed its journey north,Ed and his explorers would indeed be over exactly the same place on the Earth as at the start of the journey. Feb 7 '18 at 17:47

Inspired by @Stilez's creative answer, I've come up with an alternate solution that does not require Ed exploring alien planets. This diagram is of

the variance in the magnetic field that naturally exists on the Earth. The dashed lines represent the direction a compass would point if you were standing at that spot.

Notice that if Ed takes a heading at point ① and then follows it in a straight line South to point ② without

checking the compass until he has gone 100 units, he could end up in a situation where the compass has turned even though he has not. If he then takes a heading again at point ② going East, by the time he arrives at ③, the compass will have turned even more. It is possible that it could have turned so much that, taking a heading at point ③ for North and then — again, without rechecking the compass —

following a straight line for 100 units, Ed would end right back where he started.

Could this tricky bit of information be the answer?

Once, whilst exploring 100 units west of our starting location, me and my men headed for 100 units due south, then headed for 100 units due east and then headed for 100 units due north, and we ended up exactly where we had started.

Here's a possibility.

Ed and his men are playing an online game, while sitting together somewhere.
In the game, he and his men moved 100 units south, then 100 units east, then 100 units north - movement measured in whatever unit the game measures distance. They in fact move the distances indicated, as stated in the puzzle.
At the end of this movement, Ed and his men have moved within the game, but they are still sitting exactly where they were when they started moving.

No poles or bears were harmed, or involved, in the making of this answer.

• That was my first thought, too. But, I think the word "actually" was intended to rule this out. Feb 6 '18 at 12:05
• (@hackerb9 The longer this goes without a correct answer being acknowledged, the more it feels likely that something a bit dodgy will end up being correct hehe)
– Rubio
Feb 6 '18 at 14:55
• Or: (select to see) $\color{white}{\small{\text{moved 100 units south, then 100 units east, then 100 units north, then died in the game (surely not by bears) and were respawned in whatever place they started the game in.}}}$ (plagiarized from here). Feb 6 '18 at 16:02

# Aperture Science

What happened?

This drawing summarize it:

Let's explain it:

1. This is the starting point (green).
2. 100 units to south.
3. After walking X units east (where 0 < X < 100) they entered a portal (the yellow one in the picture, but in reality it doesn't matters which one it is).
4. The portal exit (blue) was exactly 100 units west to the entrance. So they exited there.
5. They continued walking east the remaining 100 - X units reaching the same place reached after step 2.
6. They walked north 100 units ending in the same place as step 1.

• This answer has no more merit than saying "Because magic" Feb 6 '18 at 20:52
• I think, this is a better answer than @Schrodinger'sStat thinks. Many of the other answers postulate fictional worlds, so why not a game world like Portal? Feb 6 '18 at 21:14
• @hackerb9 Apperture Science. We do what we must, because we can. For the good of all of us. Except the ones who are dead. But there is no sense crying over every mistake. You just keep on trying till you run out of cake. Feb 7 '18 at 13:25
• @Schrodinger'sStat Welcome to lateral thinking puzzles! Feb 7 '18 at 13:27

They are/were in...

... a brane/world/universe that is a square with toroidal topology while still being flat.

What that means?

Think about that games were if you exit the screen through north, you reenter the very same screen through south (and vice-versa). Ditto for east/west.

Since it is a square toroidal topology, the distance needed to go in the vertical axis until getting back to where they started is the same as in the horizontal axis. If that distance is 100 units, walking 100 units in any direction north, south, east or west (but not in any diagonal) will bring you back to were you started off. So, Ed did this three times.

Is this not walking in circles?

No, because the topology is flat. There is no circles at all. What really exists is a modulus operator working in this brane/world/universe.

So, perhaps Ed the explorer were simply...

... playing Asteroids. Ed the explorer and his men were the virtual crew of his ship.

Thanks to hackerb9 for the suggestion.

• I'm not sure that a puzzle conference on that kind of world would be rolling their eyes on questions about hypothetical spherical world's arbitrary coordinate system's behaviour around the poles of rotation being too common :-)
– Bass
Feb 6 '18 at 22:16
• So, basically, they're flatlanders exploring Asteroids. Feb 7 '18 at 5:09
• @Bass If they are visitors from another dimension or another universe, that would be quite possibly. Figureing out and determining what are the procedures and technologies needed to be performed in order to be able to visit other dimensions or other universes and come back is left as an exercise for the reader. Feb 7 '18 at 13:30
• @hackerb9 Or perhaps Ed the explorer was simply playing Asteroids. And he and his men were the virtual crew of his ship. Feb 7 '18 at 14:43
• Or they were ants exploring a donut... Feb 7 '18 at 14:49

Maybe "ended up exactly where we had started" refers to:

Time.

Notice that they

traveled 100 units south and 100 units north, taking them back to the same location. But also traveled 100 units east, possibly moving from one time zone to the next.

So :

it takes them 1 hour to complete travelling 100 units north, 100 units south and 100 units east, placing them in the next time zone where the time differs by 1 hour. So when they check the time again, it be same as when they had first started.

One other possibility relies on

magnetic drift. Currently, the magnetic pole drifts 52 km per year. They began by heading due magnetic south, then east. Then they paused until the poles drifted just enough for a due magnetic north heading to take them back to the starting point.

• That seems to have an awful lot to do with poles. Feb 6 '18 at 13:57
• @Lolgast I was hesitant to post for that reason but "heading north" or "heading south" means heading toward that pole. You can't have compass directions without poles. It at least has nothing to do with starting at, circling round, or crossing over poles. Feb 6 '18 at 14:42

Could it be:

The units are not units of space but units of time?

The OP states that they are headed in a particular direction but never actually says that they move in that direction

The Explorers could 'head' South for 100 minutes, meaning they are facing South, but not actually travelling South through space (though it could be argued that by simply waiting they were travelling through time).

Then they 'head' East for 100 minutes, again only facing East and waiting.

Then they 'head' North for 100 minutes!

They have not moved in space, but have been standing in the same spot for 5 hours.

They have however travelled the specified units heading in the specified directions

Could it be that the three 90 degree changes in direction form a triangle on the earth, therefore meaning you end up exactly where you started.

Like this, imagining that all 3 angles are 90 degrees

• Cool idea! Wouldn’t this place A at the north pole though? (The question is somewhat specific on the actual directions.)
– Bass
Feb 6 '18 at 8:36
• Exactly, this fails the explicit "no poles involved" requirement. You can't get a triangle like that without starting at the north pole. Otherwise you get something like a trapezoid with an open top. Feb 6 '18 at 15:49
• Ah, yes I hadn't thought about how the fact that specific directions are mentioned meant that pole had to be involved. I stand corrected. Feb 6 '18 at 16:47
• It should be noted that this puzzle is specifically a reference to an old canard of a puzzle where after the explorers move as described, they shoot a bear, and the question is "What color was the bear?" (The answer is "white" because they must have been at the North Pole and so it was a polar bear. Or else they were some 100+ miles north of the South Pole and so it was a "polar" bear which we assume for entertainment purposes was also white.) But this new puzzle has "no bears involved!" So your answer is 100% correct for the old canard but (by definition) must be incorrect for this new one. Feb 6 '18 at 20:21
• If it was "go 100 units, turn 90 degrees, go 100 units, turn 90 degrees, go 100 units", then you could do it away from the poles if the sphere you're on is really small. But since the directions are exactly north, south and east, I don't think it's going to work. Feb 6 '18 at 20:36

without over thinking things,

the starting location is any lattitude, especially named ones

eg

start at the Equator. go north some distance X, go East some distance Y, go south X. Check your position. Yup, its the Equator.

other possibilites could be

the tropic of cancer, the tropic of capricorn, 75'N

My answer relates to the units:

The units are an angular measure of rotation. Without loss of generality, let us assume they are grad (there are 400 grad in a circle, so 100 grad is a right angle). If the explorers stand on the spot and they simply rotate to the cardinal directions, at the end of their 'travels', they arrive at where they start (just with a different facing).

This may not fulfill the "Exactly where we had started", but maybe

The explorers began their journey on a ship heading due East.

They then left the ship heading 100 Units due south, while the ship continued on its route due East.

The explorers then traveled 100 Units due East, and from there 100 Units due North where they met back up with the ship

For this to be correct, the "same exact location" would have to refer to the ship they departed from, not geographic location.

They were lost (or maybe just in the dark).

So,

after completing their manoeuvres, they ended up exactly where they had started: Lost (or maybe just in the dark).

In his clarification, Ed specifies that they did not just go round in circles. 'Going round in circles' is a serious risk if you are, indeed, lost - so it makes sense for Ed to adopt a strategy of movement like this to avoid just wandering around in circles. The clarification may actually be intended as a veiled clue.

• The phrase "We were back exactly where we hard started", colloquially, often means that the situation has not significantly changed, even though the physical location might have done. So why the downvote? Feb 7 '18 at 13:21
• Someone just came through and carpet-bombed downvotes. Something called "tactical voting," an unfortunate and persistent part of SE. Feb 7 '18 at 15:47

Could they have been:

on a mountain? And North and South are actually Up and Down? So they were at the peak, looked around at the base of the mountain, and back to the peak?

• Orrrr maybe north was actually NW, and south was actually SW. I bet we can get about 50 more answers out of this.... Feb 7 '18 at 14:52

This might be silly, but could they be right on the International Date Line? You could make the argument that 100 units either east or west means not moving at all... You would need to identify your escape directions as Spin-wards and Counter-spin-wards, as distinct from East or West.

• East and west are synonyms for spinwards and antispinwards, and lose their meaning at the same points, i.e. the poles. The 180th meridian (which the IDL tries to follow but doesn’t always) is as far from the zero meridian as you can get, but the headings east and west definitely do exist there.
– Bass
Feb 6 '18 at 8:46

Maybe

the unit is zero. And it does not matter where they were.

• Clearer already posted an answer with that possibility before. However, nice try anyway. Feb 7 '18 at 14:22

If Ed and his men are heading for south, east and north, this assumes that they...

... know how to determine to which direction they should go in order to match those points.

There are many ways to accomplish that, one of them (a very plausible one is to) ...

... use a compass.

However, if they are carrying many metallic stuff (gets worse if there are electric currents involved) they might have...

... produced magnetic interference that confused the compass.

So what they actually called "north", "south" or "east" was in fact...

... wrong directions shown by a confused compass.

The name of the movie would be:

Around the world in 100 units
You are on a cylindrical world, or a world where it's pole to pole distance is more than twice its equatorial diameter.
On a world whose equatorial circumference is equal to 100 units you'd end up in the same position traveling east 100 units. Traveling north takes you to the starting point.

Or, more outlandishly:

Maybe I AM the planet (Also my squad are all planets) ... I front-flipped in-place 100 radians, the spun CCW for 100 radians then back-flipped in-place 100 radians. And there I am, dizzy, but right where i started. (My squad did the same PE routine)

• "[...] and didn't just go around in circles" - From the OP Feb 6 '18 at 9:16
• @Lolgast Not going around in circles means that the path is a straight line on the ground. I didn't go in circles, i went straight east. It's not my fault that my world is a cylinder Feb 6 '18 at 10:25
• That may be your interpretation, but seeing as how the OP linked other questions which mention circles in space, I'm pretty sure this was meant to be explicitly excluded. Feb 6 '18 at 10:33
• Then the question might be flawed. The use of cardinal direction is used on planets and similar 3D objects that are curved. If he said: "turn left and head straight for 100 units" my answer may be excluded but I have to disagree with OP on this. North is where your thumb points when using the right hand rule on rotating objects. East is where the closest illumination source appears above the horizon on that object. Feb 6 '18 at 11:14
• And circles are those things that appear bright as day when you view your solution outside an earth reference frame... Feb 6 '18 at 11:16