Refer to Points On A Circle.

(A) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured along the circumference, is an integer.

What is the smallest radius of the circle, given that each of the distances is unique?

(B) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured as a straight line, is an integer.

Determine the smallest radius of the circle. What is the smallest radius of the circle, given that it is rational?

Note: In Part (B) each of the distances may or may not be unique.

(In reply to re: Part A - a further reduction by Dej Mar)

I agree that in the set (1, 4, 2, 8, 9, 13, 11) the arcs (13+11) and (1+4+2+8+9) are equal but that's because they are each semicircular arcs and the set contains two points that are diametrically opposite. Surely there's no disputing that the arc length between these points is 24 which appears only once in my list of 21 distinct distances.

In fact, I believe there are 64 suitable ways of arranging 7 points around a 48 unit circumference (including reverses). 56 of these contain diametrically opposite points. Here's one that doesn't:

(1, 3, 2, 7, 10, 11, 14) 

Posted by Harry on 2010-07-15 13:26:52