Henry Ernest Dudeney/Puzzles and Curious Problems/3 - Buying Toys/Solution

by : $3$

 * Buying Toys

Solution
George bought $1$ engine, $1$ ball, $5$ dolls and $14$ trumpets.

William bought $2$ engines, $2$ balls, $1$ doll and $16$ trumpets.

Proof
A glance at the solution in the book clarifies the question, telling us that they obtained $21$ articles each.

Hence we are looking at a variant of the One Hundred Fowls problem.

Recall:
 * $1$ shilling ($20 \shillings$) is $12$ (old) pence ($12 \oldpence$)

Let all prices be expressed, therefore, in (old) pence.

Let $e$, $b$, $d$ and $t$ denote the number of engines, balls, dolls and trumpets bought respectively by either George or William.

We have:

Note that all of $e$, $b$ and $d$ need to be strictly positive.

We need to find possible values of $b$ and $d$ so as to make $27 - 5 b - 3 d$ divisible by $7$.

Let $b = 1$.

Then we have:
 * $27 - 5 - 3 d = 22 - 3 d = 7 e$

which can be satisfied (only) by $d = 5$, giving:
 * $e = 1$, $b = 1$, $d = 5$

Thus:
 * $4 e + 3 b + 2 d + \tfrac 1 2 t = 17 + \dfrac t 2 = 24$

hence making $t = 14$.

Let $b = 2$.

Then we have:
 * $27 - 10 - 3 d = 17 - 3 d = 7 e$

which can be satisfied (only) by $d = 1$, giving:
 * $e = 2$, $b = 2$, $d = 1$

Thus:
 * $4 e + 3 b + 2 d + \tfrac 1 2 t = 16 + \dfrac t 2 = 24$

hence making $t = 16$.

Let $b = 3$.

Then we have:
 * $27 - 15 - 3 d = 12 - 3 d = 7 e$

which can be satisfied by no value of $d$.

Let $b = 4$.

Then we have:
 * $27 - 20 - 3 d = 7 - 3 d = 7 e$

which can be satisfied (only) by $d = 0$.

Similarly, $b - 5$ returns no solutions.

Hence there are two possible purchases of $21$ toys for $2 \shillings$, and those are:


 * $1$ engine, $1$ ball, $5$ dolls and $14$ trumpets

and:
 * $2$ engines, $2$ balls, $1$ doll and $16$ trumpets.

The answer is completed on taking note that William bought more trumpets than George.