A Circle And Adjacent Number Puzzle

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Discussion Overview

The discussion revolves around the possibility of arranging the numbers 0 through 9 in a circle such that the difference between any two adjacent numbers is restricted to 3, 4, or 5. The conversation includes analytical reasoning and attempts to explore the implications of the arrangement conditions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that it is impossible to arrange the numbers as specified, citing constraints on the placement of certain numbers based on their neighbors.
  • Another participant suggests that a simpler explanation exists regarding the arrangement of points on a circle, arguing that only six points can fit under the given conditions, while seven are needed.
  • A later reply challenges the simplicity of the previous explanation, proposing that it is possible to place three points between two specified numbers, thus questioning the initial claim about the arrangement's impossibility.
  • Further clarification is provided regarding the condition of adjacent differences, emphasizing that the differences must be non-negative and fall within the specified range.
  • One participant acknowledges a misunderstanding of the conditions after reviewing the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of the arrangement, with some asserting it is impossible while others suggest potential arrangements that meet the conditions. The discussion remains unresolved with competing perspectives on the validity of the proposed arrangements.

Contextual Notes

Participants reference specific numerical relationships and constraints that may not be fully explored or resolved, indicating a dependence on definitions and assumptions about the arrangement conditions.

K Sengupta
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Analytically determine if it is possible to arrange the numbers 0,1,2,3,4,5,6,7,8,9 (not necessarily in this order) in a circle such that the difference between any two adjacent numbers is 3, 4 or 5.

Note: Each of the ten numbers must occur exactly once.
 
Last edited:
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Consider the loop from 9 to 9 (a sequence encompassing the other numbers).
Well, immediately after the first 9 and before the last 9 you can have only 4,5 or 6.
And after (and before) the number 1, you can have only 4,5 or 6.
So, without loss of generality, let's consider the 1 is the third number in the sequence.
Then we have
9,[4,5,6],1,[4,5,6],X,X,X,X,[4,5,6],9

Now, let's consider the numbers 0 and 8. Theirs neighbors have to be [3,4,5] but 4 and 5 are already allocated. So, the 3 has to be between 0 and 8, and this group of 3 numbes (0,3,8) has to be starting or ending the interval X,X,X,X,X. So, the remaining 2 positions are bounded by 0 or 8, which lefts no space for the 2 and 7.

So, the proposed arrangement is not possible!

:smile:
 
Rogerio, a simpler, more straightforward, explanation exists:


Arranging three points on a circle (here 3,4 and 5) creates three arcs each defined as the space between two points. For the neighboring condition to hold, an arc can only harbor two points. Hence, only six other points can be fitted onto the circle; we need seven.
 
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Werg22 said:
Rogerio, a simpler, more straightforward, explanation exists:

Arranging three points on a circle (here 3,4 and 5) creates three arcs each defined as the space between two points. For the neighboring condition to hold, an arc can only harbor two points. Hence, only six other points can be fitted onto the circle; we need seven.

How do you prove that? I don't think it is so "straightforward" as you mean...

In fact, you could place three points between the 4 and the 5: ...4,1,6,9,5...

:smile:
 
Yes, hence the subordinate clause "For the neighboring condition to hold". Placing three points as such doesn't satisfy the condition.
 
Werg22 said:
Yes, hence the subordinate clause "For the neighboring condition to hold". Placing three points as such doesn't satisfy the condition.
?!

Please, read again:

...the difference between any two adjacent numbers is 3, 4 or 5.
THIS is the only neighboring condition.
And the difference between any two adjacent numbers from the sequence "4,1,6,9,5" IS 3, 4 or 5!

(BTW: the "difference between any two adjacent numbers" is always a non negative number...)

:smile:
 
I see, I had understood something else, sorry.
 
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