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The proof is easier than you might think. There are 5 odd numbers (1, 3, 5, 7, 9) in the range, but only four even numbers (2, 4, 6, 8). There are not, consequently, enough even numbers to separate all the odd numbers. Two odd numbers must inevitably appear next to each other somewhere in the circle, totalling a non-prime even number. Odd and even numbers must alternate all the way around a sum circle if it is to have prime values. This will only work with an even number of places in the circle.

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