This one is easy: start with the unit square in R2 and a point in the center. Move the center point up in the Z direction until it reaches any convenient rational or integer value. --- Wrt Means, Gosper's objection was a bit more subtle: should we require that avg( avg(a,b), avg(c,d) ) = same thing with b and c swapped? This *is* true for ordinary averages, and geometric, harmonic, etc. means. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of Daniel Asimov Sent: Friday, April 29, 2005 3:59 PM To: math-fun Subject: Re: [math-fun] Triangles question I heard this question in the immediately equivalent version: < Is there a point in the plane at rational distances from each corner of a unit square?

My former thesis advisor, Moe Hirsch, has remarked, "If we don't know that, we don't know *anything*!" Here's a wider class of questions: Suppose we have a planar unit square in R^n for any n >= 2. Then is there a point of R^n at rational distance from each corner of the square? And if that fails: Suppose we have a planar unit square in Hilbert space H = {f: Z+ -> R : sum of all f(k)^2 < oo }. Then is there a point of H at rational distance from each corner of the square? (Here d(f,g) = sum of all (f(k)-g(k))^2.) --Dan ----------------------------------------------------------------- Richard Guy writes: << What we'd REALLY like to know is: Is there a point at integer distances from each corner of a square with integer edge?

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