Henry Ernest Dudeney/Puzzles and Curious Problems/20 - Boys and Girls/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $20$

Boys and Girls
Nine boys and three girls agreed to share equally their pocket-money.
every boy gave an equal sum to every girl,
and every girl gave another equal sum to every boy.
Every child then possessed exactly the same amount.
What was the smallest possible amount that each then possessed?


Solution

Let us assume that the smallest unit of money is $1$.

On that basis, each boy possessed $12$ and he gave $1$ to each girl.

Every girl then had $36$, and gave $3$ to each boy.

Everybody then possessed $18$.

Martin Gardner's repackaging of this puzzle in 536 Puzzles & Curious Problems was answered on the basis of that unit being $1$ cent.


In England at the time this puzzle was set, the smallest unit of money was $\tfrac 1 4 \oldpence$

On that basis, each boy possessed $3 \oldpence$ and he gave $\tfrac 1 4 \oldpence$ to each girl.

Every girl then had $9 \oldpence$ and gave $\tfrac 3 4 \oldpence$ to each boy.

Everybody then possessed $4 \tfrac 1 2 \oldpence$


Proof


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