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        riddles >> medium >> Irrational punching/painting (revisited)
        
(Message started by: Aryabhatta on Jun 13^th, 2008, 10:24am)

Title: Irrational punching/painting (revisited)
Post by Aryabhatta on Jun 13^th, 2008, 10:24am

You have a device which is capable of painting all points at an irrational distance from a given point (in the 2D plane).

To use the device, you just have to select that point and all points at an irrational distance from that point are painted black.

Initially you start out with all points white in the 2D plane.

Prove/Disprove:

To paint the whole plane black, selecting the three points (0,0), (1,0) and (sqrt(2),0) is enough.

Title: Re: Irrational punching/painting (revisited)
Post by TenaliRaman on Jun 13^th, 2008, 2:00pm

I am not for one to get any kind of solutions, but I can sure find links ;D

Irrational Coloring (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1148278845)
Irrational Device (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1190953268)
Non-integer colouring (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1148507742) (Not same but thematically similar)

-- AI

Title: Re: Irrational punching/painting (revisited)
Post by Aryabhatta on Jun 13^th, 2008, 2:21pm

I did check two of those threads and decided to use the word "revisited" because those threads already exist.

I believe the probem stated above hasn't been treated in any of those threads, but, they are all too long for me to be sure.

Title: Re: Irrational punching/painting (revisited)
Post by Eigenray on Jun 13^th, 2008, 3:28pm

No I don't think this problem was considered. It was shown that (0,0), (1,0), and (t,0) suffice whenever t is not algebraic of degree 1 or 2 over the rationals. Obviously that doesn't apply here.

[hide]But it looks like t=sqrt(2) still works. Otherwise we would find positive integers with a⁴ - b⁴ = c², which has no nontrivial solutions by descent.[/hide]

I wonder, can we describe all t for which (0,0), (1,0), and (t,0) do not work?

Title: Re: Irrational punching/painting (revisited)
Post by Aryabhatta on Jun 13^th, 2008, 3:35pm

Yes, that is exactly what I had in mind, Eigenray!

The other problem you state is interesting...

Title: Re: Irrational punching/painting (revisited)
Post by Aryabhatta on Jun 13^th, 2008, 4:21pm

t = sqrt(17) does not work. (0,4/3) is at a rational distance from (0,0), (1,0), (sqrt(17), 0).

Title: Re: Irrational punching/painting (revisited)
Post by Eigenray on Jun 13^th, 2008, 7:06pm

There will be many such t. For example, if t² = (a/b)² - (2mn/(m²-n²))², with a,b,m,n integers.

More generally, if r,s are rationals with r+s>1, |r-s|<1, then we can pick a point (x,y) a distance r from (0,0) and s from (1,0). Then (t,0) can be a point of intersection of the x-axis with any circle about (x,y) with sufficiently large rational radius. This leads to a parameterization with 3 rational parameters. But I don't know if there's a simpler form. In particular, is there an algorithm for deciding whether a given number works?