Henry Ernest Dudeney/Modern Puzzles/27 - The First-Born's Legacy/Solution

by : $27$

 * The First-Born's Legacy

Solution
Freddy gets $\pounds 800$.

performs the arithmetic that determines the exact sums to be bestowed:


 * Mrs. Goodheart gets $\dfrac {43} {63} \times \pounds 8400 = \pounds 5333 \tfrac 1 3 = \pounds 5333 \ 6 \shillings 8 \oldpence$


 * the first-born gets $\dfrac {12} {63} \times \pounds 8400 = \pounds 1600$


 * the second-born gets $\dfrac 6 {63} \times \pounds 8400 = \pounds 800$


 * his brother gets $\dfrac 5 {63} \times \pounds 8400 = \pounds 666 \tfrac 2 3 = \pounds 666 \ 13 \shillings 4 \oldpence$

He then goes on to say:
 * But the sting is in the tail, and the question is indeterminate.

What is "indeterminate" is what is meant by "his brother".

goes on to explain that the twins were born either side of the time at which the clock went back at the end of British Summer Time.

Hence Tommy was actually the first-born and Freddy the younger of the two.

A pencil-written marginal note in the second-hand copy possessed by prime.mover says:
 * Silly

with which prime.mover concurs.

Proof
We first note that $\dfrac 2 3 + \dfrac 1 5 + \dfrac 1 {10} + \dfrac 1 {12} = \dfrac {40 + 12 + 6 + 5} {60} = \dfrac {63} {60}$.

As this is greater than $1$, we must interpret the question as that the legacies are in the appropriate proportions.

Hence:
 * Mrs. Goodheart gets $\dfrac {43} {63} \times \pounds 8400 = \pounds 5333 \tfrac 1 3 = \pounds 5333 \ 6 \shillings 8 \oldpence$


 * the first-born gets $\dfrac {12} {63} \times \pounds 8400 = \pounds 1600$


 * the second-born gets $\dfrac 6 {63} \times \pounds 8400 = \pounds 800$


 * his brother gets $\dfrac 5 {63} \times \pounds 8400 = \pounds 666 \tfrac 2 3 = \pounds 666 \ 13 \shillings 4 \oldpence$

According to the question, it appears that in fact Freddy is the first-born.

However, pulls a new fact out of his hat:
 * Tommy was actually born when the clock showed just before $2$ a.m. on $21$st September $1924$
 * At $2$ a.m. the clocks went back $1$ hour, as British Summer Time ended at that point
 * Freddy was born soon after $1$ a.m. by the clock, but before $2$ a.m.

Hence Tommy is actually the first-born, and not Freddy.

So Freddy gets $\pounds 800$.