Henry Ernest Dudeney/Puzzles and Curious Problems/135 - Distributing Nuts/Solution

by : $135$

 * Distributing Nuts

Solution
Aunt Martha kept $321$ nuts for herself.

Proof
Let $n$ be the total number of nuts.

Let $a$, $b$, $c$, $d$ and $m$ be the number of nuts received or retained by Tommy, Bessie, Bob, Jessie and Aunt Martha respectively.

We have:

Rewriting $(1)$ to $(5)$ in a more convenient form:

At this stage we recognise that $a$, $b$, $c$ and $c$ must be positive integers, which gives us the obvious Ansatz:

We note that:
 * $a + c = 100$
 * $b + d = 75$

giving a difference between $a + c$ and $b + d$ of $25$.

So we multiply our Ansatz by $4$ to get:

We see that:

giving the appropriate difference.

Thus: