Clifford Pickover gave several sets of circles in his Grand Internet Math Challenge, with the object being to find consecutive integers that made them magic circles. One set of circles (Figure 4.34 in Pickover's book) has the properties that allow for a simple assignment of integers to get a magic circle. Here's one such setting that assigns the values 1..48 to the intersection points, giving circles totals of 294:

Note that there are eight total circles, each of which intersects six of the other circles in two places, and that exactly two circles meet at each intersection point. Further, there are twelve intersection points on each circle. If we search, we can find four additional circles that pass through eight intersection points:

With the numbers available, we have the potential to assign the values 1..48 in such a way as to obtain two more circles that that sum to the magic value of 294. However, looking more closely, you will find that we can not make such an assignment to the middle two of the dark circles above, and if we make such an assignment to either the inner or outer dark circle, then the values at the other circle are forced. Thus, we can assign values to get only one more circle that sums to 294, either the inner dark circle or the outer dark circle. Here's an assignment that makes the outer dark circle sum to 294:

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