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Re: Two distances among Four Points

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On 20/03/2026 17:11, James Dow Allen wrote:
> 
> "Carl G." <carlgnews@microprizes.com> posted:
> 
>> On 3/17/2026 12:14 PM, James Dow Allen wrote:
>>> How many ways are there to arrange 4 points in the plane such that
>>> only two inter-point distances occur?
> 
>> For me, the hardest of the six was the second arrangement that does not
>> include an equilateral triangle.
> 
> And it is Carl G. who successfully completes the sextet.
> This solution -- which Mr. G carefully leaves unspoiled -- is surely
> the hardest to come up with.

Indeed.  I see the solution now.  It's the only one whose geometric compass-and-straightedge 
construction isn't obvious.  (Euclid knew how to do it, but I doubt your average school leaver would 
get very far!)

Mike.

> 
> Now:  Solve the companion puzzle:
> James Dow Allen <user4353@newsgrouper.org.invalid> posted:
>> A closely related puzzle is to arrange FIVE distinct points such that
>> only two inter-point distances occur.
> 
> ... and describe the weird/wonderful relationship between this 5-point
> companion puzzle and the question I asked earlier:
>> Which of the six {4-point, 2-distances} arrangements is hardest to think of?
> 

I think this 5-point puzzle is likely to be easier for many people than finding the 6'th solution to 
the 4-point puzzle, and might lead them to the 6'th solution indirectly!  (That's amusing when you 
think about it...)

Anyway, nice puzzle!

Mike.

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Thread

Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-17 19:14 +0000
  Re: Two distances among Four Points Charlie Roberts <croberts@gmail.com> - 2026-03-18 16:36 -0400
    Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-20 06:03 +0000
      Re: Two distances among Four Points  Charlie Roberts <croberts@gmail.com> - 2026-03-20 12:50 -0400
    Re: Two distances among Four Points Charlie Roberts <croberts@gmail.com> - 2026-03-20 11:04 -0400
      Re: Two distances among Four Points Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-03-20 16:22 +0000
        Re: Two distances among Four Points Phil Carmody <pc+usenet@asdf.org> - 2026-05-19 20:01 +0300
  Re: Two distances among Four Points "Carl G." <carlgnews@microprizes.com> - 2026-03-20 09:58 -0700
    Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-20 17:11 +0000
      Re: Two distances among Four Points Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-03-21 15:55 +0000
        Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-22 07:20 +0000
          Re: Two distances among Four Points HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> - 2026-03-23 16:16 +0000
            Re: Two distances among Four Points Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-03-25 02:40 +0000
              Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-25 11:44 +0000
                Re: Two distances among Four Points "Carl G." <carlgnews@microprizes.com> - 2026-03-25 09:17 -0700
                Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-26 07:58 +0000
                Re: Two distances among Four Points "Carl G." <carlgnews@microprizes.com> - 2026-03-26 10:26 -0700
                Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-27 10:25 +0000
                Re: Two distances among Four Points James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-03-27 20:27 +0000

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