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Problem of the Week
Problem E
Stand in a Circle

The numbers from \(1\) to \(17\) are arranged around a circle. One such arrangement is shown.

The seventeen numbers are arranged like the numbers on an analogue clock, but placed in a seemingly random order: 13, 4, 14, 10, 5, and so on.

Explain why every possible arrangement of these numbers around a circle must have at least one group of three adjacent numbers whose sum is at least \(27\).

In solving the above problem, it may be helpful to use the fact that the sum of the first \(n\) positive integers is equal to \(\tfrac{n(n+1)}{2}\). That is, \[1 + 2 + 3 + … + n = \frac{n(n+1)}{2}\]