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Points On A Circle (Posted on 2004-08-14) Difficulty: 4 of 5
I placed six points on the circumference of a circle such that the distance between any two of the points is an integer. What is the smallest such circle I could use?

What if each distance must be unique?

No Solution Yet Submitted by ThoughtProvoker    
Rating: 4.4286 (7 votes)

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Solution First part (Solution) | Comment 7 of 17 |

If the distances (straight lines, not curves) don't have to be distinct, then the smallest circle appears to be one of radius 12.5 units, with the six points as follows (centered on the origin of the xy-axes):

Point 1: (12.5 , 0)
Point 2: (10.54 , 6.72)
Point 3: (-3.5 , 12)
Point 4: (-12.5 , 0)
Point 5: (-10.54 , -6.72)
Point 6: (3.5, -12)
 
Distances:
Points 1 to 2:  7
Points 1 to 3:  20
Points 1 to 4:  25
Points 1 to 5:  24
Points 1 to 6:  15
Points 2 to 3:  15
Points 2 to 4:  24
Points 2 to 5:  25
Points 2 to 6:  20
Points 3 to 4:  15
Points 3 to 5:  20
Points 3 to 6:  25
Points 4 to 5:  7
Points 4 to 6:  20
Points 5 to 6:  15
 
 

  Posted by Penny on 2004-08-15 21:39:33
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