A Six-Pointed Star 

    a
b  c d  e
 f     g
h  i j  k
    l
  1. Place the numbers 1 to 12 in the 12 locations on the six-pointed star, such that each of the six lines of four numbers has the same sum and the sum of the six points of the star has the same sum as the lines.
  2. Can you find an arrangement for which the six lines and the set of six star-points produce seven consecutive sums?

Source: 1. Henry Ernest Dudeny, 536 Curious Problems & Puzzles, #394. 2. Original.

Solutions were received from Denis Borris, Philippe Fondanaiche, Larry Baum, and Joseph DeVincentis. There are 6 possible solutions for the first problem and 499 for the second problem. Larry Baum had a computer do the hard part for him after some brief analysis: 
1) Since each letter is used twice and 1+2+...+12 = 78, all six lines
total 2*278 => each line = 26
We can assume A is the smallest of the values of any of the points of
the star and that B < E (otherwise rotate and flip if needed).
With the restriction on the sum of the 6 points = 26 there are 6
different solutions:

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

2) Let the consecutive numbers be: x-3, x-2, x-1, x, x+1, x+2, x+3.  One
of these is the sum of the points.  Depending on which one that is, the
other 6 add up to some number between 6x-3 and 6x+3.  But they still
must add to 2*278 which is a multiple of 6; so x = 26 and the 6 lines
total 23, 24, 25, 27, 28 and 29.
There are 499 possible solutions!

(1 5 8 7 9 10 12 6 11 4 3 2)
(1 5 7 8 9 11 12 6 10 4 3 2)
[... 495 other solutions removed - KD ...]
(1 2 10 12 3 9 7 4 6 8 5 11)
(1 2 10 12 3 8 6 4 7 9 5 11)
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