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Family ages (Posted on 2019-04-16)

Three families make a remarkable discovery. The sum of the ages of their members are all the same, the sum of the squares of the ages of their members are all the same, and the sum of the cubes of the ages of their members are all the same. Everyone in all 3 families has a different age, and nobody is more than 100 years old.

What is the smallest possible sum of their ages? Can this be done with 4 families?

No Solution Yet Submitted by Danish Ahmed Khan

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soln to first part

Comment 4 of 4 |

Three satisfying families have ages:

(14 25 11 12 13), (19 23 2 6 10 15), (20 22 3 5 9 16)

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.

The code is here:

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

Edited on April 20, 2019, 7:04 am

Posted by Steven Lord on 2019-04-19 16:37:45

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