## Ken's Puzzle of the WeekEight Magic Lines

Place the numbers 1-16 in locations A-P such that each line has the same sum.  Solve for all possible sums.

Source: Original (though likely elsewhere, too...)
Solutions were received from Philippe Fondanaiche, Denis Borris, K Sengupta, and Dan Chirica.  Philippe's analysis below matches most submissions.  Dan Chirica discovered that this diagram doesn't work for just any set of sequential numbers; the largest set is 9-24, with a common sum of 77.  No higher set of sequential numbers will work.

Let S the common sum to the eight lines abd a,b,c,d,e,f,...the values in locations A,B,C,D,E,F,...
We have 8 relations such as a + m + p + f = S, a + i + n + j + d = S, b + n + o + e = S, etc...
By addition of these relations, we can infer 2*(a + b + c + d +....) + m + n + o + p = 8S.
As a + b + c + d +....+ o + p = 17*16/2 = 136, we get S = 34 + (m + n + o + p )/8. That means that the sum of the 4 terms m, n, o and p must be a multiple of 8.
On the other hand by eliminating successively a,b,c,d...in the eight relations, we get a relation between the remaining variables i, j, k, l, m, n, o and p : 2*( i + j + k + l) = m + n + o + p.
As the minimum value of i + j + k + l is 1 + 2 + 3 + 4 = 10, the minimum value of m + n + o + p is >=2*10 = 20. The fisrt number >= 20 divisible by 8 is 24. So the minimum value of S  is 37.
At last the maximum value of of m + n + o + p is 16 + 15 + 14 + 13 = 58. The first number <=58 which is divisible by 8 is 56. So the maximum value of S is 41.
As a conclusion the possible values of S are 37, 38, 39, 40 and 41.
There exist many solutions for each of these values.
Hereafter, one solution is given as an example for each possible value of S.

 Sum = 37 Sum = 38 14 15 14 15 3 8 12 4 5 16 37 16 2 7 13 38 6 2 4 3 9 8 7 13 37 9 12 11 6 38 1 1 37 11 10 37 38 10 5 38 37 37 37 37 38 38 38 38 Sum = 39 Sum = 40 9 16 11 10 1 8 13 4 14 8 39 7 4 15 14 40 6 10 12 1 12 15 7 5 39 6 16 13 5 40 3 3 39 11 2 39 40 9 2 40 39 39 39 39 40 40 40 40 Sum = 41 13 7 9 4 11 14 12 41 6 3 8 16 15 2 41 10 41 1 5 41 41 41 41 41

Mail to Ken