Enigma 1700: Polymagic
From New Scientist #2867, 2nd June 2012 [link]
From a box of counters numbered 1 to 12, Joe asked Penny to select a set of six and place them on the corners of a regular hexagon and then place the remaining six counters on the mid-points of each side, so that the number on each of these counters was the sum of the numbers on the counters on the two adjacent corners. Joe then produced 16 counters and asked Penny to repeat the challenge, but this time selecting a set of eight and placing them, and the remaining eight, similarly on an octagon.
In the first case, Penny found her choice of the set of six counters was rather limited. How many choices did she have?
And how many choices did she have in the case of the octagon?