Ken's Puzzle of the Week

Magic Snowflake

Put the numbers 1-13 at the corners of the 6-pointed snowflake to make each diamond have the same sum.  Minimize the common sum.

To compare answers, rotate & reflect your solution to put the smallest number the top, and the next smallest clockwise.

Extension (added 6Feb2009): Find solutions for all possible sums.

Source: Original.

Solutions were received from Kirk Bresniker, Dan Chirica, Mark Rickert, Joseph DeVencentis, and Bernie R. Erickson.  The minimum sum is 21.  There are three solutions.  Here is Joe's summary: 
      5
10  8   7   9
  2   1   4
12  6   3  13
     11

      5
11  7   8   9
  2   1   3
12  6   4  13
     10

      7
13  3  10   8
  4   1   2
11  5   6  12
      9
To achieve this sum, you must have 1 in the center and one of these sets on the inner points: 2 3 4 6 7 8, 2 3 4 5 7 9, 2 3 4 5 6 10.
For the first set, 5 can only be in a diamond with 7 and 8, 13 with 3 and 4, and 12 with 2 and 6. So these pairs of inner numbers are adjacent, and only the first two solutions shown above work for the remaining diamonds.
For the second set, 6 can only be on a diamond with 9 and 5, 12 with 5 and 3. But 8 can only go with either 7 and 5, or 9 and 3, and neither of these pairs is possible with the other diamonds
For the third set, 7 must go with 10 and 3, 8 with 10 and 2, and 9 with 5 and 6. 11 can go with 4 and 5 or with 3 and 6, but if you put it with 3 and 6, there is no solution.  The third solution above is the only way to complete the snowflake.
 

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