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 Spacy colors (Posted on 2006-08-25)
Every point in 3D-space is colored either red, green or blue. Let R (resp. G and B) be the set of distances between red (resp. green and blue) points. Prove that at least one of R, G, or B, consists of all the non-negative real numbers.

 See The Solution Submitted by JLo Rating: 4.0000 (2 votes)

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 Solution (Some details required) | Comment 5 of 10 |
` `
`NOTATION:`
`  R* = { P in E^3 | P is colored red }  G* = { P in E^3 | P is colored green }  B* = { P in E^3 | P is colored blue }`
`  R = { d(P,Q) | P,Q in R* }   G = { d(P,Q) | P,Q in G* }   B = { d(P,Q) | P,Q in B* }`
`  X << Y denotes X is a subset of Y`
`  X ^ Y denotes the intersection of sets X and Y`
`  X v Y denotes the union of sets Y and Y`
`  X - Y  denotes the set difference of sets X and Y`
`  sphere(X,y) denotes the sphere with center X              and radius y `
`PROOF (Others can flesh out the details):`
`  Assume r in [0,inf) - R,         g in [0,inf) - G, and         b in [0,inf) - B`
`  WOLOG let 0 < b <= g <= r.`
`  Pick a point L in R*. Then sphere(L,r) << G* v B*.`
`  Case A: sphere(L,r) << G*`
`          Pick a point M in sphere(L,r). Then`
`          sphere(M,g) ^ sphere(L,r) << G*`
`          This is a contradiction since it is a           space circle containing colorless points.`
`  Case B: sphere(L,r) << B*`
`          Pick a point M in sphere(L,r). Then`
`          sphere(M,b) ^ sphere(L,r) << B*`
`          This is a contradiction since it is a           space circle containing colorless points.`
`  Case C: (sphere(L,r) ^ B*) and (sphere(L,r) ^ G*)          are both non-empty                     Pick a point M in sphere(L,r) ^ G*. Then`
`          sphere(M,g) ^ sphere(L,r) << B*. Pick a`
`          point N in sphere(M,g) ^ sphere(L,r). Then`
`          sphere(N,b) ^ sphere(M,g) ^ sphere(L,r) << B*`
`          This is a contradiction since it is a set           containing two colorless points.`
`  Therefore, at least one of the sets R, G, or B must  equal [0,inf).`
` Oops - Back slashes don't work to well.`
`To clean up the details:`
`Case 1: 0 < b <= g <= r`
` This is the case discribed above.`
`Case 2: 0 <= g <= r`
` This case is described in Case A above.`
`Case 3: 0 <= 0 <= r`
` No points colored green or blue. Clearly,`
` R = [0,inf)`
`Case 4: 0 <= 0 <= 0`
` All points are colorless. Contradicts problem statement.`

Edited on August 25, 2006, 9:41 pm

Edited on August 25, 2006, 9:49 pm

Edited on August 27, 2006, 11:54 am
 Posted by Bractals on 2006-08-25 17:23:07

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