1997 PMWC Problems/Problem T10

Problem

The twelve integers $1, 2, 3,\dots, 12$ are arranged in a circle such that the difference of any two adjacent numbers is either $2, 3,$ or $4$ . What is the maximum number of the difference $4$ can occur in any such arrangement?

Solution

The answer is $\boxed{0}.$

We first seat $1.$ Then we put all odd numbers in one way and all even numbers in another direction. The result is $12,10,8,6,4,2,1,3,5,7,9,11,12...$ when we read the table. It satisfies the conditions and never has $4$ as a difference.

1997 PMWC (Problems)
Preceded by Problem T9	Followed by Last Problem
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10

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1997 PMWC Problems/Problem T10

Problem

Solution

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