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Scrutiny (Posted on 2023-05-01) Difficulty: 2 of 5
A puzzle by V. Dubrovsky, from Quantum, January-February 1992:

In a certain planetary system, no two planets are separated by the same distance. On each planet sits an astronomer who observes the planet closest to hers.

Prove that if the total number of planets is odd, there must be a planet that no one is observing.

See The Solution Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

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soln | Comment 1 of 6
We can categorize the planets as binaries or field planets. 
Binaries come in pairs; they are each other's closest neighbors. 
A field planet is a planet that has a closest neighbor that
has a different closest neighbor. 

We populate the system with planets with successively 
increasing "nearest neighbor distances" and the first two
planets introduced make a binary pair by definition. If all 
the planets turn out to be in binary pairs, then we 
have an even number of planets, all looking at each other,

With an odd number of planets, if we require every planet be 
observed, we are in trouble. 

With an odd number, we must have at least one field planet. 
The problem with having any field planet is that although 
it could be looking at a member of a binary pair, or at
another field planet, in order to be observed, it requires 
a _different_ field  planet to be looking at it, and _that_ 
field planet then requires _yet another_ field planet to be 
looking at _it_, and so on, ad infinitum.  

So, with an odd number of planets, there must be at least one 
field planet and thus at least one unobserved field planet.

(As an aside: Even with an even number of planets, there may 
be one or more field planets, and even one of these is one too many.)   

Edited on May 3, 2023, 3:36 am
  Posted by Steven Lord on 2023-05-01 16:34:21

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