Clifford Pickover gave several sets of circles in his Grand Internet Math Challenge, with the object being to find consecutive integers that make them magic circles. The following is a proof that there is no magic solution using consecutive, positive integers greater than 0 for one of the set of circles in his challenge. The set of rings is the following:

I have colored and labeled these rings to assist in my proof.

The proof starts by showing that there is no assignment of the integers
*1,...,28* to the intersection points that results in a set of magic
rings. Assume there is such an assignment, and that the sum is *k*.
Consider the red circle. The points labeled *A _{i}*
and the points labeled

Likewise, if we consider the blue circle and the green circle, we see that

Eliminating *k* from Equations 1 and 2, and eliminating
*k* from Equations 1 and 3 gives us:

Likewise, a similar derivation shows that

Summing these four equations and keeping only one copy of each *L _{i}*
on the right gives us the inequality

Now, we will maximize the sum of the values on the left-hand size of Equation 4 and minimize the sum of the values on the right-hand side of Equation 4.

To maximize the sum of *A _{1}*,

Thus, we see that the maximum of the left-hand-side of Equation 4 is
less than the minimum of the right-hand-side of the equation, and
hence, no assignment of the integers *1,...,28* to these nodes will
result in a magic circle.

From here it is easy to show that no assignment of consecutive
integers starting with *j>1* will result in a magic circle. Again
consider Equation 4, which must hold for any assignment of
integers. We know that assigning the integers *1,...,28* will
result in the left hand side being strictly less than the right-hand
side. If we add *j-1* to each integer in such an assignment, the
right-hand size increases by *18(j-1)* while the left hand side
increases by only *6(j-1)*, and the inequality is worsened.

This proof, of course, says nothing about using a mix of positive and
negative integers, nor does it say anything about using non-consecutive
integers.

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