with summed: ages, sq. ages, cubed ages = 75, 1255, 23625
An alternate second family: (17 24 1 8 10 15) also works.
I have not found a four family solution, but have decided to not continue searching until a clearer definition of "family" is made.
For the solution above, I used the constraints I mentioned in my first post but limited the number of children to be <=4.
program s66
implicit none
integer i1,i2,i3,i4,i5,i6,j1,j2,j3,j4,j5,j6,
1 jjj1,jjj2,jjj3,jjj4,jjj5,jjj6,
2 sumi,sumii,sumiii,sumj,sumjj,sumjjj,j,cnt,dcnt
cnt=0
dcnt=0
do 12 i1=14,100
c print*,'i1 = ',i1
do 11 i2=i1+1,100
do 10 i3=0,i1-1
if(i3.ge.i1.or.i3.ge.i2)go to 10
do 9 i4=0,i1-1
if(i4.ge.i1.or.i4.ge.i2)go to 9
if(i3+i4.ne.0.and.i4.le.i3)go to 9
do 8 i5=0,i1-1
if(i5.ge.i1.or.i5.ge.i2)go to 8
if(i3+i4+i5.ne.0.and.(i5.le.i4.or.i5.le.i3))go to 8
do 7 i6=i5+1,i1-1
if(i6.ge.i1.or.i6.ge.i2)go to 7
if(i3+i4+i5+i6.ne.0.and.
1 (i6.le.i5.or.i6.le.i4.or.i6.le.i3))go to 7
sumi= i1 +i2 +i3 +i4 +i5 +i6
sumii= i1**2+i2**2+i3**2+i4**2+i5**2+i6**2
sumiii=i1**3+i2**3+i3**3+i4**3+i5**3+i6**3
do 6 j1=14,100
c print*,'j1= ',j1
j=j1
jjj1=j**3
if(jjj1.gt.sumiii) go to 7
if(j.eq.i1.or.j.eq.i2.or.j.eq.i3
1.or.j.eq.i4.or.j.eq.i5.or.j.eq.i6)go to 6
do 5 j2=j1+1,100
j=j2
jjj2=j**3
if(jjj1+jjj2.gt.sumiii) go to 7
if(j.eq.i1.or.j.eq.i2.or.j.eq.i3
1 .or.j.eq.i4.or.j.eq.i5.or.j.eq.i6)go to 5
do 4 j3=0,j1-1
j=j3
jjj3=j**3
if(jjj1+jjj2+jjj3.gt.sumiii) go to 7
if(j.ge.j1.or.j.ge.j2)go to 4
if(j.ne.0.and.(j.eq.i1.or.j.eq.i2.or.j.eq.i3
1 .or.j.eq.i4.or.j.eq.i5.or.j.eq.i6))go to 4
do 3 j4=0,j1-1
j=j4
jjj4=j**3
if(jjj1+jjj2+jjj3+jjj4.gt.sumiii) go to 7
if(j.ge.j1.or.j.ge.j2)go to 3
if(j.ne.0.and.(j.eq.i1.or.j.eq.i2.or.j.eq.i3
1 .or.j.eq.i4.or.j.eq.i5.or.j.eq.i6))go to 3
if(j3+j4.ne.0.and.j4.le.j3)go to 3
do 2 j5=0,j1-1
j=j5
jjj5=j**3
if(jjj1+jjj2+jjj3+jjj4+jjj5.gt.sumiii) go to 7
if(j.ge.j1.or.j.ge.j2)go to 2
if(j.ne.0.and.(j.eq.i1.or.j.eq.i2.or.j.eq.i3
1 .or.j.eq.i4.or.j.eq.i5.or.j.eq.i6))go to 2
if(j3+j4+j5.ne.0.and.(j5.le.j4.or.j5.le.j3))go to 2
do 1 j6=j5+1,j1-1
j=j6
jjj6=j**3
sumjjj=jjj1+jjj2+jjj3+jjj4+jjj5+jjj6
if(sumjjj.gt.sumiii) go to 7
if(j.ge.j1.or.j.ge.j2)go to 1
if(j.ne.0.and.(j.eq.i1.or.j.eq.i2.or.j.eq.i3
1 .or.j.eq.i4.or.j.eq.i5.or.j.eq.i6))go to 1
if(j3+j4+j5+j6.ne.0.and.
1 (j6.le.j5.or.j6.le.j4.or.j6.le.j3))go to 1
cnt=cnt+1
if(cnt.gt.1000000000)then
cnt=0
dcnt=dcnt+1
print 100,dcnt,i1,i2,i3,i4,i5,i6,j1,j2,j3,j4,j5,j6,sumiii,sumjjj
100 format(i6,3x,2(6(i3,1x),6x)2(i10,1x))
endif
sumj= j1 +j2 +j3 +j4 +j5 +j6
if(sumi.ne.sumj)goto 1
sumjj= j1**2+j2**2+j3**2+j4**2+j5**2+j6**2
if(sumii.ne.sumjj)go to 1
if(sumiii.ne.sumjjj)go to 1
print 101,i1,i2,i3,i4,i5,i6,j1,j2,j3,j4,j5,j6
101 format('soln ',3x,2(6(i3,1x),6x))
print*,sumi,sumii,sumiii
1 enddo
2 enddo
3 enddo
4 enddo
5 enddo
6 enddo
7 enddo
8 enddo
9 enddo
10 enddo
11 enddo
12 enddo
end
The winning rows were the "B" rows below.
soln 16 20 5 9 10 15 14 21 4 11 12 13 A
75 1087 17325
soln 17 19 6 8 9 16 14 21 4 11 12 13 A
75 1087 17325
soln 17 22 1 6 11 12 14 23 0 9 10 13
69 1075 18837
soln 17 24 1 8 10 15 14 25 0 11 12 13 B
75 1255 23625
soln 17 26 0 3 4 15 14 27 0 0 11 13
65 1215 25955
soln 17 26 3 7 9 16 14 27 1 11 12 13 C
78 1360 27684
soln 18 19 6 7 8 17 14 22 3 11 12 13 D
75 1123 18675
soln 18 20 5 7 10 15 14 22 3 11 12 13 D
75 1123 18675
soln 18 22 0 8 9 11 14 23 0 6 12 13
68 1074 19052
soln 18 24 4 5 6 17 14 26 0 9 12 13
74 1266 24974
soln 18 25 0 2 4 11 14 26 0 0 7 13
60 1090 22860
soln 18 29 5 6 7 16 14 30 1 11 12 13 E
81 1531 35001
soln 18 30 4 5 7 16 14 31 0 10 12 13
80 1570 37460
soln 19 21 4 6 8 17 14 24 1 11 12 13 F
75 1207 21825
soln 19 23 2 6 10 15 14 25 0 11 12 13 B
75 1255 23625
soln 19 28 0 2 5 9 14 29 0 0 7 13
63 1255 29673
soln 20 21 4 5 7 18 14 25 0 11 12 13 B
75 1255 23625
soln 20 21 4 5 10 15 14 24 1 11 12 13 F
75 1207 21825
soln 20 22 3 5 9 16 14 25 0 11 12 13 B
75 1255 23625
soln 20 25 5 6 7 15 14 27 1 11 12 13 C
78 1360 27684
soln 20 30 0 5 7 10 14 31 0 2 12 13
72 1474 36468
soln 20 30 1 7 8 9 14 31 0 5 12 13
75 1495 36585
soln 20 31 0 2 8 9 14 32 0 0 11 13
70 1510 39040
soln 21 34 0 5 7 8 14 35 0 1 12 13
75 1735 49545
soln 21 35 0 4 6 8 14 36 0 0 11 13
74 1782 52928
soln 21 36 1 4 5 6 14 37 0 0 9 13
73 1815 56323
soln 22 28 0 5 6 9 14 30 0 1 12 13
70 1410 33670
soln 24 28 0 5 6 7 14 31 0 0 12 13
70 1470 36460
soln 25 26 6 7 8 9 14 30 1 11 12 13 E
81 1531 35001