by Guest » Wed Jun 25, 2014 2:12 pm
Let
[tex]p[/tex] be the age of the parent
[tex]m[/tex] be the age of the middle child
[tex]d[/tex] the difference in ages between successive children
The question states [tex]p^2 = (m-4d)^2+(m-3d)^2+(m-2d)^2+(m-d)^2+m^2+(m+d)^2+(m+2d)^2+(m+3d)^2+(m+4d)^2[/tex]
which expands and cancels to
[tex]p^2 = 9m^2 + 60d^2[/tex]
Clearly the right hand side is a multiple of [tex]3[/tex] which implies [tex]p^2[/tex] is a multiple of [tex]3[/tex] and therefore [tex]p[/tex] is a multiple of [tex]3[/tex]. Let [tex]p = 3t[/tex], so now we have
[tex]9t^2= 9m^2 + 60d^2[/tex]
which rearranges to
[tex](t+m)(t-m) = 20d^2/3[/tex]
The left hand side is an integer which implies [tex]d^2[/tex] and therefore [tex]d[/tex] is a multiple of [tex]3[/tex].
In order for the youngest child to be born we must have [tex]m-4d\geq 0[/tex] which implies [tex]m\geq 4d[/tex] and so
[tex]p^2\geq 9(4d)^2+60d^2 = 204d^2 \geq (14d)^2[/tex]
[tex]p \geq 14d[/tex]
We know [tex]d[/tex] is a multiple of [tex]3[/tex], so [tex]d[/tex] must be [tex]0, 3 ,[/tex] or [tex]6[/tex], any higher and [tex]p[/tex] would be at least [tex]126[/tex] years old which is unreasonable.
Case 1: d = 0
There are lots of solutions if we allow the children to be nonuplets for example [tex]p=30, m=10,[/tex] however I don't think there is a documented case of nonuplets surviving for more than a year after birth (however octuplets are known to have survived). This is an unlikely solution.
Case 2: d = 3
From our previous work we know
[tex](t+m)(t-m) = 60[/tex]
We also know that [tex]p = 3t>m+12[/tex] (parent is older than the oldest child) and [tex]m\geq 12[/tex] (youngest child is at least 0), this means that
[tex]3t>24[/tex] which implies [tex]t>8[/tex] and [tex]t+m>20[/tex]
Since [tex]t+m[/tex] and [tex]t-m[/tex] are factors of [tex]60[/tex] that leaves us with two possibilities
[tex]t+m = 60, t-m = 1[/tex] (which fails as this implies [tex]t = 30.5[/tex])
or
[tex]t+m = 30, t-m = 2[/tex] which implies [tex]t = 16, m= 14[/tex].
So the age of the parent [tex]p = 48[/tex], the youngest child is [tex]2[/tex], the oldest is [tex]26[/tex] and the interval is [tex]3[/tex] years.
Case 3: d = 6
We proceed similarly to the previous case.
From our previous work we know
[tex](t+m)(t-m) = 240[/tex]
We also know that [tex]p = 3t>m+24[/tex] (parent is older than the oldest child) and [tex]m\geq 24[/tex] (youngest child is at least 0), this means that
[tex]3t>48[/tex] which implies [tex]t>16[/tex] and [tex]t+m>40[/tex]
Since [tex]t+m[/tex] and [tex]t-m[/tex] are factors of [tex]240[/tex] that leaves us with several possibilities:
[tex](t+m,t-m) =[/tex]
[tex](48,5)[/tex] [fails as [tex]t[/tex] is not an integer]
[tex](60,4)[/tex]
[tex](80,3)[/tex] [fails as [tex]t[/tex] is not an integer]
[tex](120,2)[/tex] [fails as [tex]t=61[/tex] implies the parent's age is [tex]183[/tex]]
[tex](240,1)[/tex] [fails as [tex]t[/tex] is not an integer]
This leaves us with
[tex]t+m = 60, t-m = 4[/tex] which implies [tex]t = 32, m= 28[/tex].
So the age of the parent [tex]p = 96,[/tex] the youngest child is [tex]4[/tex], the oldest is [tex]52[/tex] and the interval is [tex]6[/tex] years. This is an unlikely solution but theoretically possible.
Hope this helped,
R. Baber.