Stamps
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Stamps
You get a magic stamp that hits all points of the area that has transcendental distance from the origin.
How often do you need to stamp to cover the entire plane?
Extending to an ndimensional stamp, how often do you have to stamp to cover the entire R^n?
How often do you need to stamp to cover the entire plane?
Extending to an ndimensional stamp, how often do you have to stamp to cover the entire R^n?

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Re: Stamps
For the plane you need two stamps: (0,pi) and (pi,0), this will cover all of the points. This is not a rigorous proof feel free to prove me wrong!
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
Re: Stamps
tomtom2357 wrote:For the plane you need two stamps: (0,pi) and (pi,0), this will cover all of the points.
Sadly, this isn't quite enough.
Spoiler:
Spoiler:
Spoiler:

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 Joined: Tue Jul 27, 2010 8:48 am UTC
Re: Stamps
Okay, I guess I need to look harder next time. Thanks for the solution though!
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
Re: Stamps
An explicit solution:
Edit:
Spoiler:
Edit:
Spoiler:
Re: Stamps
Nitrodon wrote:Edit:Spoiler:
That does not work.
Spoiler:
 skeptical scientist
 closedminded spiritualist
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Re: Stamps
mfb wrote:That does not work.Spoiler:
To elaborate on this and sfwc's earlier comment:
sfwc wrote:In R^n:Spoiler:
Spoiler:
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Stamps
mfb wrote:That does not work.Spoiler:
I think it does.
Spoiler:
skeptical scientist wrote:Spoiler:
Spoiler:
 skeptical scientist
 closedminded spiritualist
 Posts: 6142
 Joined: Tue Nov 28, 2006 6:09 am UTC
 Location: San Francisco
Re: Stamps
Nitrodon wrote:Spoiler:
Damn, missed that.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Stamps
Oh, good point.
And a stamp which covers all algebraic distances does not work (with countable stamp usage), so there is no obvious followup problem .
And a stamp which covers all algebraic distances does not work (with countable stamp usage), so there is no obvious followup problem .

 Posts: 563
 Joined: Tue Jul 27, 2010 8:48 am UTC
Re: Stamps
How about this: for ndimensions, stamping at point (x_{1},x_{2},x_{3},...) covers all points (y_{1},y_{2},y_{3},...) such that x_{i}y_{i} is algebraic for exactly one value of i. You can generalize this to x_{i}y_{i} is algebraic for exactly n values of i
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
Re: Stamps
I would expect that it is not possible to cover the plane (or higher dimensions) with this. Take the plane and assume that a finite number N of stamps covers all points. From these N, let n be stamps where the first coordinate (x) difference is algebraic, and at Nn stamps the second coordinate (y) difference is algebraic. There are coordinates X which have transcendental difference in x to all xpositions of the first group, therefore they are not covered by the first stamp type.
With the same argument, there are coordinates Y which have transcendental difference in y to all ypositions of the second group, therefore they are not covered.
Now, take an xvalue out of X and an yvalue out one out of Y and you get a point which is not covered.
I hope I did not make a mistake here. If that argument is not flawed, the only way to get a coverage is "the coordinate difference is algebraic for exactly 0 values".
With the same argument, there are coordinates Y which have transcendental difference in y to all ypositions of the second group, therefore they are not covered.
Now, take an xvalue out of X and an yvalue out one out of Y and you get a point which is not covered.
I hope I did not make a mistake here. If that argument is not flawed, the only way to get a coverage is "the coordinate difference is algebraic for exactly 0 values".
Re: Stamps
Qaanol wrote:Nitrodon is correct.Spoiler:
That's certainly the most elegant argument I've seen for this. Nice!
My own is certainly far less elegant:
Spoiler:
I fell for the same "sloppy" argument that the number of points would increase with dimensions at first as well, which was, I thought, a nice twist to the classical (2dimensional) problem. Its first occurance known to me was in the usenet group rec.puzzles and in fact can still be found in the FAQ there (e.g. here, then with irrational stamps).
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