Henry Ernest Dudeney/Puzzles and Curious Problems/4 - Puzzling Legacies/Solution

by : $4$

 * Puzzling Legacies
 * A man bequeathed a sum of money, a little less than $\pounds 1500$, to be divided as follows:
 * The five children and the lawyer received such sums that
 * the square root of the eldest son's share,
 * the second son's share divided by two,
 * the third son's share minus $\pounds 2$,
 * the fourth son's share plus $\pounds 2$,
 * the daughter's share multiplied by two,
 * and the square of the lawyer's fee
 * all worked out at exactly the same sum of money.


 * No pounds were divided, and no money was left over after the division.


 * What was the total amount bequeathed?

Solution

 * The oldest son gets $\pounds 1296$.
 * The second son gets $\pounds 72$.
 * The third son gets $\pounds 38$.
 * The fourth son gets $\pounds 34$.
 * The daughter gets $\pounds 18$.
 * The lawyer's fee is $\pounds 6$.

Hence the total amount bequeathed is $\pounds 1464$.

Proof
Let $a$, $b$, $c$, $d$, $e$ and $l$ be the shares received by the eldest son, second son, third son, fourth son, daughter and lawyer respectively.

We have:

We can either attempt to solve this quartic formally, or try some integer values of $l$.

There are not many such that $l^4 < 1500$, so this is a feasible approach.

and we have gone far enough.

$5$ is immediately eliminated because $\dfrac 9 2 5^2$ is not an integer.

$4$ looks too small for the numbers under discussion.

So, we try $6$:


 * $6^4 + \dfrac 9 2 6^2 + 6 = 1296 + 162 + 6 = 1464$

which fits the bill perfectly.

The result follows.