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Points On A Circle II (Posted on 2010-07-13) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Some Thoughts Upper and Lower bounds (part A) | Comment 1 of 12
I assume the distance between any pair of points in part A is measured along the shorter path between them.

Lower bound:
Since there are 7*6/2 = 21 pairs, the greatest pairwise distance must be at least 21, (assuming that it is the pairwise distances that must be unique).  That means that the total circumference must be at least 21*2 = 42.

Upper bound:  

One obvious method (presumably not optimal) is to have the distance between adjacent points be 1,2,4,8,16,32 and 64. This gives a circumference of 127. 

So, the circumference in part A is between 42 and 127, with a corresponding (irrational) radius, assuming my two interpretations of the puzzle.  

Edited on July 13, 2010, 1:00 pm

Edited on July 13, 2010, 1:15 pm
  Posted by Steve Herman on 2010-07-13 12:50:46

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