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

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Puzzles and Curious Problems by Henry Ernest Dudeney: $3$

Buying Toys
George and William were sent out to buy toys for the family Christmas tree,
and, unknown to each other, both went at different times to the same little shop,
where they had sold all their stock of small toys
except engines at $4 \oldpence$, balls at $3 \oldpence$ each, dolls at $2 \oldpence$ each, and trumpets at $\tfrac 1 2 \oldpence$ each.
They both bought some of all, and obtained $21$ articles, spending $2 \shillings$ each.
But William bought more trumpets than George.
What were their purchases?


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:

\(\text {(1)}: \quad\) \(\ds 4 e + 3 b + 2 d + \tfrac 1 2 t\) \(=\) \(\ds 24\) the amount spent
\(\text {(2)}: \quad\) \(\ds e + b + d + t\) \(=\) \(\ds 21\) the total number of toys
\(\ds \leadsto \ \ \) \(\ds 7 e + 5 b + 3 d\) \(=\) \(\ds 27\) $2 \times (1) - (2)$
\(\ds \leadsto \ \ \) \(\ds 7 e\) \(=\) \(\ds 27 - 5 b - 3 d\)

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.

$\blacksquare$


Sources

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