I developed this within a spreadsheet finding only one solution. I have copied Penny's list below the "--o-----o----" line. This solution is a variant of her sixth one where Z equals 42. Note too, in her comments she suggests there are multiple arrangements for each case.
A B C D E F G H I J K L M N O
13 8 2 6 15 10 11 4 1 5 7 14 9 3 12
Ring Values:
1. A B F G 13,8,11,10 [42]
2. B C D G H I J K 8,1,4,2,5,1,7,6 [44] (Black)
3. D E K L 14,7,6,15 [42]
4. F G H I M N 11,10,9,4,5,3 [42]
5. I J K L N O 5,3,1,12,14,7 [42]
----o-----o-----o-----o-----o-----o-----o-----o-----o-----o
A=15 B=7 C=8 D=9 E=14 F=13 G=2 H=3 I=1 J=5 K=4 L=10 M=12 N=6 O=11
[A+B+F+G] = [15+7+13+2] = 37
[B+C+D+G+H+I+J+K] = [7+8+9+2+3+1+5+4] = 39
[D+E+K+L] = [9+14+4+10] = 37
[F+G+H+I+M+N] = [13+2+3+1+12+6] = 37
[I+J+K+L+N+O] = [1+5+4+10+6+11] = 37
----------------------------------------
A=15 B=4 C=8 D=9 E=14 F=13 G=6 H=2 I=1 J=7 K=3 L=12 M=11 N=5 O=10
[A+B+F+G] = [15+4+13+6] = 38
[B+C+D+G+H+I+J+K] = [4+8+9+6+2+1+7+3] = 40
[D+E+K+L] = [9+14+3+12] = 38
[F+G+H+I+M+N] = [13+6+2+1+11+5] = 38
[I+J+K+L+N+O] = [1+7+3+12+5+10] = 38
----------------------------------------
A=14 B=3 C=7 D=8 E=15 F=13 G=9 H=2 I=1 J=6 K=5 L=11 M=10 N=4 O=12
[A+B+F+G] = [14+3+13+9] = 39
[B+C+D+G+H+I+J+K] = [3+7+8+9+2+1+6+5] = 41
[D+E+K+L] = [8+15+5+11] = 39
[F+G+H+I+M+N] = [13+9+2+1+10+4] = 39
[I+J+K+L+N+O] = [1+6+5+11+4+12] = 39
----------------------------------------
A=15 B=2 C=6 D=8 E=14 F=13 G=10 H=3 I=1 J=5 K=7 L=11 M=9 N=4 O=12
[A+B+F+G] = [15+2+13+10] = 40
[B+C+D+G+H+I+J+K] = [2+6+8+10+3+1+5+7] = 42
[D+E+K+L] = [8+14+7+11] = 40
[F+G+H+I+M+N] = [13+10+3+1+9+4] = 40
[I+J+K+L+N+O] = [1+5+7+11+4+12] = 40
----------------------------------------
A=15 B=2 C=4 D=9 E=14 F=13 G=11 H=3 I=1 J=7 K=6 L=12 M=8 N=5 O=10
[A+B+F+G] = [15+2+13+11] = 41
[B+C+D+G+H+I+J+K] = [2+4+9+11+3+1+7+6] = 43
[D+E+K+L] = [9+14+6+12] = 41
[F+G+H+I+M+N] = [13+11+3+1+8+5] = 41
[I+J+K+L+N+O] = [1+7+6+12+5+10] = 41
----------------------------------------
A=15 B=2 C=3 D=9 E=13 F=14 G=11 H=4 I=1 J=6 K=8 L=12 M=7 N=5 O=10
[A+B+F+G] = [15+2+14+11] = 42
[B+C+D+G+H+I+J+K] = [2+3+9+11+4+1+6+8] = 44
[D+E+K+L] = [9+13+8+12] = 42
[F+G+H+I+M+N] = [14+11+4+1+7+5] = 42
[I+J+K+L+N+O] = [1+6+8+12+5+10] = 42
----------------------------------------
A=15 B=3 C=1 D=10 E=12 F=14 G=11 H=4 I=2 J=6 K=8 L=13 M=7 N=5 O=9
[A+B+F+G] = [15+3+14+11] = 43
[B+C+D+G+H+I+J+K] = [3+1+10+11+4+2+6+8] = 45
[D+E+K+L] = [10+12+8+13] = 43
[F+G+H+I+M+N] = [14+11+4+2+7+5] = 43
[I+J+K+L+N+O] = [2+6+8+13+5+9] = 43
----------------------------------------
A=14 B=4 C=1 D=9 E=12 F=15 G=11 H=2 I=3 J=6 K=10 L=13 M=8 N=5 O=7
[A+B+F+G] = [14+4+15+11] = 44
[B+C+D+G+H+I+J+K] = [4+1+9+11+2+3+6+10] = 46
[D+E+K+L] = [9+12+10+13] = 44
[F+G+H+I+M+N] = [15+11+2+3+8+5] = 44
[I+J+K+L+N+O] = [3+6+10+13+5+7] = 44
----------------------------------------
A=14 B=6 C=1 D=9 E=11 F=15 G=10 H=2 I=3 J=4 K=12 L=13 M=7 N=8 O=5
[A+B+F+G] = [14+6+15+10] = 45
[B+C+D+G+H+I+J+K] = [6+1+9+10+2+3+4+12] = 47
[D+E+K+L] = [9+11+12+13] = 45
[F+G+H+I+M+N] = [15+10+2+3+7+8] = 45
[I+J+K+L+N+O] = [3+4+12+13+8+5] = 45
----------------------------------------
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