Way Too Many Points on The Line
Is there a set of natural numbers such that every natural number (except zero) appears as the (absolute) difference of exactly one pair of them?
Also try: A Harder version of the Problem
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2 solutions
@Agnishom Chattopadhyay ,is it correct?
The set { 1 , 2 , 4 , 8 , . . . } is a possible solution. Indeed, for each n > = 0 2 n = 2 n + 1 − 2 n .
Which pair gives a difference of 5 ?
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Indeed, I misunderstood the problem. I understood it as follows : is there a set of natural numbers such that each number in the set is the difference of exactly one pair of them ?
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So any ideas on how to form the desired set of natural numbers?
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@Brian Charlesworth – I can’t find a simple formula, but I figured a repeatable construction method :
1 - choose any starting number m (this is the first number of the set)
2 - find the smallest difference s that is still not managed with the actual set
3 - add s to the largest number of the set M , that gives M ’ (don’t remove M from the set)
4 - add M ’ to the set
5 - update the list of the differences you have managed with this new number M ’ (pair it with all the other numbers in the set)
6 - repeat step 2 to 6
I will later write a python script that follows this algorithm.
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@Matthieu Courbaize – Ok, great. That's the same method I was considering, but I wasn't sure if that was what Agnishom had in mind as a proof.
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@Brian Charlesworth – That is exactly what I had in mind. Maybe you can post this as a solution?
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@Agnishom Chattopadhyay – Actually it doesn’t work either, here are the first numbers of the set starting with 0 : { 0 , 1 , 3 , 7 , 1 2 , 2 0 , 3 0 , 4 4 , 5 9 , 7 5 , . . . }
The difference of 29 is done twice : by 3 0 − 1 and by 5 9 − 3 0 ...
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@Matthieu Courbaize – Okay, this is interesting. Maybe I have not really thought this through. This puzzle seems quiet interesting nonetheless.
I was excited to see such a direct solution. But alas, this does not work.
The set of all natural numbers {1,2,3,4.......}