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Hello Randy,

Hope you can help solve this conundrum sent to you from the U.K.

Peter has nine children all born at regular intervals. The sum of the square of their ages, which are all exact years, is equal to the square of his own age. So how old is he?

Many thank for your time and my best wishes to you and yours for the festive season and beyond.

Bill

In order to completely solve this problem, we need to know a) the time between the children's birthday, n, b) the current age of the youngest child, C c) the age father when the first child was born, Ai, and the current age of the father, Ac At any rate, here's a partial solution;

Sum of squares of children's age = ∑(j=0->8)(C + jn)^2 = (Ai + Ac)^2 = square of father's age.

Expanding the terms in the sum

9C^2 +2Cn∑(j=0->8)j + n^2∑(j=0->8)j^2.

The symbol ∑ means the sum of the terms indexed by j. To evaluate the expression, we need 2 results for the sum of arithmetic series. First is

∑(j=1->m)aj = (m/2)(a1 +am), which for our case becomes (9/2)(0 + 8) = 36. We also need

∑(j=1->m)j^2 = ( m(m+1)(2m+1) )/6 which becomes ( 8(8+1)(2･8+1)/6 = 204. Putting it all together gives

(Ai + Ac)f^2 = 9C^2 + 2Cn(36) + n^2(204).

As can be seen, we need the 3 parameters listed above to compute an actual number.

Hope this helps.

Randy

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Comment | Many thanks, Randy. It certainly is difficult and fraught with lack of info. Warm regards Bill |

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