## Interlocking Circles

1.  ``` ____ ____ / \ / \ A X B / / \ \ | D E | | |_F_| | | /| |\ | \ G H I J / \ / X \ / X__K_/ \_L__X | | \ / \ / \ / --C-- ``` Three interlocking circles create seven distinct regions and divide the three circles into 12 segments (labels A-L on the figure). Find A-L as distinct positive integers such that each of the seven regions has the same sum (when adding the three values surrounding each region.) Try to find the numbers such that the maximum of (A..L) is as small as possible. Repeat the previous problem with the added constraint that the sum around each of the three circles should also be the same (different than that of the seven regions, of course.) If possible, for comparison purposes, please submit your diagrams or solutions such that A is your maximum value, and B is your minimum value for that A. (You should be able to reassign numbers in your solution to do this.)
2. The centers of the three circles create an equilateral triangle. If we assume the side-length of this triangle is 1, what is the radius of the circles (all identical) which makes the lengths of segments D and I the same?

Source: Original.

Solution
Mail to Ken