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