Two Coins Math Problem

Here is a little problem that has a usually unexpected answer. But I'm sure you'll get it right.
When will we be able to read 500 again?
Enjoy this one!

[DacuberTM]

if both coins are equal, after half a turn it will be 500 again.

[Jivvi]

Think about what part of each coin would be in contact with the other. They are the same size, so by the time the coin that starts at the top has gone half way round the other one (to the bottom), half of its circumference has been in contact with the edge of the other coin, so the point that is now touching is opposite the point that was originally touching. It started with the bottom edge touching, and it now has the top edge touching, and it is below the other coin, so it is right side up.

[HLEBR]

I believe it´s half a turn because if it were rolling over a flat surface, then it would take the coin a full 360° turn in order to return to its original position. But given that it rolls over a circumference that I asume is the same size as the coin, and starts moving at its highest point, then when it rolled only 180° you should actually add another 180° due to the reflection it suffers  it reaches the bottom part of the circumference. Also, an important condition is that the coin doesn´t slip as it turns. 

[zysis]

180 because the coin is flipped essentially, the bottom is now in contact with the top. You assume because it rolls that it does a straight line, thus giving you the idea it would be a 360, but because it rolls on the other coin, it's a 180.

[mamlasu]

@ilovemylefthandedcig That is incorrect. The center of rotation of the moving coin isn't the center of the idle coin, but it is the point where the coins touch. This point itself rotates around the center of the idle coin as the moving coin moves, so you get double rotation. Just take 2 coins and make sure.

[TyYann]

You don't change the order of the digits, you just rotate the coin.

[vikram.kannada]

1/2 turn.
When two radii are the same, the percentage of rotation = percentage of revolution at any point of time. At 1/2 turn, the coin rotates 50% and hence the 500 rotates by 50% with respect to the centre-to-circumference direction. This means, it is straight when seen from circumference to-centre direction

[БранимирМилошевић]

Since they are the same size coin, as you travel around the circumference of the "inner" coin, you are essentially "peeling" the circumference of the "outer" coin as the exact same rate, i.e., you can think of the trace of the "outer" coin's circumference being converted into linear distance as it moves.

Going by that, it seems like the answer should be one rotation (360 degrees).

Can someone tell me if I messed up?

[aquaabib]

One turn. Since the two coins have the same perimeter, the only way the bottom of the 500 is going to touch the other coin is when the full perimeter has been developed, which is on the top where it all started.

[psychicalligator]

Half a turn, because at 1/4 it is upside-down

[Driulinkin]

I believe that the rotation of the coin proceeds along with its orbiting around the other one. So, after 1 orbit, it's exactly the same as before...

[TyYann]

@CrazyChainSawLuigi Both coins are identical.

[SkyKing]

I haven't tried it yet but if the circumference of both is the same it should be at one full time around. That makes sense to me.

[WanderingTimelord]

Neat little problem. Not a difficult solution. The motion of the top coin around the bottom is effectively equivalent to the rotation of the bottom coin (if both had fixed centers), so extra distance is traveled. I guess that might not be completely intuitive, though.

[DoubleOrKnot]

at 180º the coin should be right side up, because the moving coin would normally be upside down at that point except it's moving around the other coin, so you flip it, making it right side up again ^.^ ... that was fun.

[taurbeer]

The next time the 500 is lined up correctly is as far as I remember after the coin has been turned 114, 59155902616464175359630962821 degrees.

[CrazyChainSawLuigi]

that's simple. you figure the circumference of both coins then you figure what whole number times the circumference of the other coin equals a multiple of the circumference of the 500 coin

correct me if I'm wrong

[NicolasBadworstKuche]

if you would turn it horizontal line it would take 1 turn.So why in this case it just takes 1/2 turn?
The problem is that you have to add the rotation of the unmoved coin. What i mean with that is, if you would move a pin on the surface of the unmoved coin, it would be upside down on the bottom side. =1/2 turn
That means if the rotation of the turning coin + the rotation of the unmoved coin =1 you got a full turn of the moving coin.
I hope you know what i mean, and sry for my bad english ;)

[krestella]

after 1 round ( after turning the coin into 360 degree it will return to it same position )

[GarrettandAndrew]

half turn because you can read it clearly upside down and that is what you asked to see it and read it clearly if your saying to read it back to its original upwards position it will take a full turn because it is imposible to see it like that for a quarter turn or a pie turn that is my solution