The maximum life span of a given inhabitant of the planet Realmblanc is 950 years (by Earth standards.)

The following relationships hold among the ages of the members of a family of four in Realmblanc. Each of the ages (in Earth years) is a positive integer.

(a) The mother is three times as old as the daughter was when the father was the same age as the mother is now.

(b) When the daughter reaches half the age the mother is now, the son will be half as old as the father was when the mother was twice the age the daughter is now.

(c) When the father reaches twice the age the mother was when the daughter was the same age as the son is now, the daughter will be four times as old as the son is now.

(d) The sum of the ages of the son and the mother (in Earth years) is divisible by 7.

Given that precisely one of their ages (in Earth years) is a perfect square, what are the four ages?

F = Father, M = Mother, D = Daughter, S = Son The three equations are as follows: M = 3[D - (F - M)] [(M/2 - D) + S] * 2 = F - (M - 2D) {[M - (D - S)] * 2 - F} + D = 4S

F = Father, M = Mother, D = Daughter, S = Son

The three equations are as follows:

M = 3[D - (F - M)]
[(M/2 - D) + S] * 2 = F - (M - 2D)
{[M - (D - S)] * 2 - F} + D = 4S

Simplifying these equations and loading them in a matrix yields this:

<TABLE border=1 cellSpacing=5 cellPadding=5 width="100%"> <TBODY> <TR> <TH>F</TH> <TH>M</TH> <TH>D</TH> <TH>S</TH> <TH>tot</TH></TR> <TR> <TD width="20%">-1</TD> <TD>2</TD> <TD>-1</TD> <TD>-2</TD> <TD>0</TD></TR> <TR> <TD>-1</TD> <TD>2</TD> <TD>-4</TD> <TD>2</TD> <TD>0</TD></TR> <TR> <TD>3</TD> <TD>-2</TD> <TD>-3</TD> <TD>0</TD> <TD>0</TD></TR></TBODY></TABLE>

Row reduction will yield:
<TABLE border=1 cellSpacing=5 cellPadding=5 width="100%"> <TBODY> <TR> <TH>F</TH> <TH>M</TH> <TH>D</TH> <TH>S</TH> <TH>tot</TH></TR> <TR> <TD width="20%">1</TD> <TD>0</TD> <TD>0</TD> <TD>-11/3</TD> <TD>0</TD></TR> <TR> <TD>0</TD> <TD>1</TD> <TD>0</TD> <TD>-7/2</TD> <TD>0</TD></TR> <TR> <TD>0</TD> <TD>0</TD> <TD>1</TD> <TD>-4/3</TD> <TD>0</TD></TR></TBODY></TABLE>

At first glance, it is clear that the Son's age must be divisible by 2 & 3.
It is also clear that the Mother's age is already divisible by 7. Because the sum M + S must be divisible by 7, S must also be divisible by 7.
Therefore, S is a multiple of 42.

Counting by 42's, the Mother's age reaches 21 as S = 126.

F = 462
M = 441
D = 168
S = 126

Posted by hoodat on 2009-12-03 17:40:27